Consider the following construction:
We can go in reverse, and start constructing such examples by starting with a number, and inserting digits into it. If we get a prime, we can continue to add more digits. If not, we can try to add the digit at another place.
Edit: If we are also considering leading zeroes that get erased after truncating the first digit, we get few extra examples that weren't included in the previous construction.
I've found all of the examples for first few digit cases ;
Or to be more specific, if we look at such numbers with
$d=2,3,4,5,6\dots$ digits, we have:
$$20 ,118 ,734 ,4679, 31722\dots$$
such prime numbers among $d$ digit numbers.
It would seem that this sequence would continue growing implying that there are infinitely many such prime numbers. But this does not need to be the case.
You can see the lists of all examples split into lists based on their length (digits) :
$d = 2$
[11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 97]
$d = 3$
[101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 401, 419, 421, 431, 433, 439, 443, 457, 461, 463, 467, 479, 487, 491, 503, 509, 523, 541, 547, 563, 569, 571, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 733, 739, 743, 751, 761, 769, 773, 797, 811, 823, 829, 839, 853, 859, 863, 883, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 997]
$d = 4$
[1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1123, 1151, 1153, 1163, 1181, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1277, 1279, 1283, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1427, 1429, 1433, 1439, 1451, 1459, 1481, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1549, 1567, 1571, 1579, 1597, 1601, 1607, 1609, 1613, 1619, 1627, 1637, 1657, 1663, 1667, 1693, 1697, 1699, 1709, 1723, 1733, 1753, 1759, 1783, 1789, 1801, 1811, 1823, 1831, 1861, 1867, 1871, 1873, 1879, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2029, 2039, 2053, 2063, 2069, 2083, 2111, 2113, 2129, 2131, 2137, 2141, 2161, 2179, 2203, 2213, 2237, 2239, 2243, 2269, 2273, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2371, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2441, 2467, 2503, 2539, 2593, 2609, 2617, 2633, 2647, 2659, 2663, 2671, 2677, 2683, 2689, 2693, 2699, 2711, 2713, 2719, 2729, 2731, 2741, 2791, 2797, 2803, 2833, 2837, 2843, 2903, 2939, 2953, 2963, 2969, 2971, 3001, 3011, 3019, 3023, 3037, 3041, 3061, 3067, 3079, 3083, 3109, 3119, 3121, 3137, 3163, 3167, 3181, 3187, 3191, 3217, 3229, 3253, 3259, 3271, 3301, 3307, 3313, 3319, 3331, 3347, 3359, 3361, 3371, 3373, 3391, 3407, 3413, 3433, 3457, 3461, 3463, 3467, 3491, 3511, 3517, 3529, 3533, 3539, 3541, 3547, 3559, 3571, 3583, 3593, 3607, 3613, 3617, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3823, 3833, 3847, 3853, 3863, 3907, 3911, 3917, 3919, 3929, 3931, 3947, 3967, 4001, 4003, 4007, 4013, 4019, 4021, 4051, 4057, 4073, 4079, 4091, 4127, 4129, 4133, 4139, 4157, 4159, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4261, 4271, 4283, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4421, 4423, 4457, 4463, 4483, 4493, 4507, 4517, 4519, 4523, 4547, 4561, 4567, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4651, 4657, 4663, 4673, 4679, 4691, 4721, 4729, 4733, 4751, 4759, 4787, 4789, 4793, 4799, 4801, 4817, 4831, 4861, 4871, 4877, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4987, 5003, 5009, 5011, 5023, 5039, 5059, 5099, 5101, 5107, 5113, 5147, 5167, 5171, 5179, 5197, 5209, 5231, 5233, 5237, 5273, 5303, 5309, 5323, 5347, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5471, 5477, 5479, 5503, 5563, 5569, 5623, 5639, 5641, 5647, 5653, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5741, 5743, 5791, 5839, 5869, 5903, 5923, 5939, 5953, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6091, 6101, 6113, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6211, 6217, 6229, 6247, 6263, 6269, 6271, 6277, 6301, 6311, 6317, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6397, 6421, 6427, 6451, 6473, 6481, 6491, 6529, 6547, 6553, 6563, 6569, 6571, 6577, 6599, 6607, 6619, 6653, 6659, 6661, 6673, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6791, 6793, 6803, 6823, 6829, 6833, 6841, 6863, 6883, 6907, 6911, 6917, 6947, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7039, 7043, 7069, 7079, 7103, 7109, 7127, 7129, 7151, 7159, 7193, 7211, 7219, 7229, 7243, 7283, 7297, 7307, 7309, 7331, 7333, 7349, 7351, 7369, 7393, 7433, 7451, 7457, 7487, 7517, 7523, 7541, 7547, 7561, 7573, 7591, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7691, 7699, 7703, 7723, 7753, 7793, 7823, 7829, 7853, 7873, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7951, 8011, 8017, 8039, 8053, 8059, 8101, 8111, 8117, 8123, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8389, 8419, 8423, 8429, 8431, 8443, 8461, 8467, 8513, 8537, 8539, 8543, 8563, 8573, 8597, 8599, 8623, 8629, 8641, 8647, 8663, 8677, 8693, 8719, 8753, 8761, 8783, 8803, 8831, 8837, 8839, 8863, 8893, 8923, 8929, 8941, 8963, 8971, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9059, 9067, 9103, 9109, 9127, 9137, 9151, 9157, 9161, 9173, 9181, 9199, 9209, 9239, 9241, 9277, 9283, 9293, 9311, 9319, 9337, 9341, 9371, 9377, 9397, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9533, 9539, 9547, 9601, 9613, 9619, 9629, 9631, 9643, 9661, 9677, 9679, 9697, 9719, 9721, 9733, 9739, 9743, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9829, 9833, 9839, 9859, 9871, 9883, 9907, 9929, 9941, 9967, 9973]
I can upload lists for $d\ge5$ if you want.
This does not answer your question as for an actual proof (disproof), this is still an open problem as Barry Cipra wrote in the comments.
Michael in his answer provides a reasoning of why this conjecture is likely to be true.