Your inequality
$$(\sigma(n))^2\leq \sigma_0(n)\sigma_2(n)$$
is the Cauchy-Schwarz inequality in $\mathbb R^{\sigma_0(n)}.$
One of the vectors used in Cauchy Schwarz has all entries equal to $1,$ the other has all the divisors of $n$ as the entries.
Spanish Primorial
I tried it for the primorials, it would appear that there is no upper bound, even when dividing by your $H_n.$
m 2 = 2 ratio 1.11111 ratio / log m 1.60299
m 6 = 2 3 ratio 1.38889 ratio / log m 0.775154
m 30 = 2 3 5 ratio 2.00617 ratio / log m 0.589843
m 210 = 2 3 5 7 ratio 3.13465 ratio / log m 0.586232
m 2310 = 2 3 5 7 11 ratio 5.31148 ratio / log m 0.685795
m 30030 = 2 3 5 7 11 13 ratio 9.2138 ratio / log m 0.89368
m 510510 = 2 3 5 7 11 13 17 ratio 16.4938 ratio / log m 1.25494
m 9699690 = 2 3 5 7 11 13 17 19 ratio 29.8538 ratio / log m 1.8557
m 223092870 = 2 3 5 7 11 13 17 19 23 ratio 54.9393 ratio / log m 2.85799
m 6469693230 = 2 3 5 7 11 13 17 19 23 29 ratio 102.798 ratio / log m 4.5505
m 200560490130 = 2 3 5 7 11 13 17 19 23 29 31 ratio 193.147 ratio / log m 7.42177
m 7420738134810 = 2 3 5 7 11 13 17 19 23 29 31 37 ratio 366.498 ratio / log m 12.3669
m 304250263527210 = 2 3 5 7 11 13 17 19 23 29 31 37 41 ratio 698.922 ratio / log m 20.9579
m 13082761331670030 = 2 3 5 7 11 13 17 19 23 29 31 37 41 43 ratio 1335.75 ratio / log m 35.9943
m 614889782588491410 = 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ratio 2562.51 ratio / log m 62.5609
m 32589158477190044730 = 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 ratio 4938.71 ratio / log m 109.919
m 1922760350154212639070 = 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ratio 9553.67 ratio / log m 194.941
m 117288381359406970983270 = 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 ratio 18500.9 ratio / log m 348.292
Here is the same computer run, but i have saved space by not printing m or its factorization, just the largest prime dividing m.
biggest prime factor 2 ratio 1.11111 ratio / log m 1.60299
biggest prime factor 3 ratio 1.38889 ratio / log m 0.775154
biggest prime factor 5 ratio 2.00617 ratio / log m 0.589843
biggest prime factor 7 ratio 3.13465 ratio / log m 0.586232
biggest prime factor 11 ratio 5.31148 ratio / log m 0.685795
biggest prime factor 13 ratio 9.2138 ratio / log m 0.89368
biggest prime factor 17 ratio 16.4938 ratio / log m 1.25494
biggest prime factor 19 ratio 29.8538 ratio / log m 1.8557
biggest prime factor 23 ratio 54.9393 ratio / log m 2.85799
biggest prime factor 29 ratio 102.798 ratio / log m 4.5505
biggest prime factor 31 ratio 193.147 ratio / log m 7.42177
biggest prime factor 37 ratio 366.498 ratio / log m 12.3669
biggest prime factor 41 ratio 698.922 ratio / log m 20.9579
biggest prime factor 43 ratio 1335.75 ratio / log m 35.9943
biggest prime factor 47 ratio 2562.51 ratio / log m 62.5609
biggest prime factor 53 ratio 4938.71 ratio / log m 109.919
biggest prime factor 59 ratio 9553.67 ratio / log m 194.941
biggest prime factor 61 ratio 18500.9 ratio / log m 348.292
biggest prime factor 67 ratio 35929.5 ratio / log m 626.784
biggest prime factor 71 ratio 69890.7 ratio / log m 1134.84
biggest prime factor 73 ratio 136055 ratio / log m 2065.29
biggest prime factor 79 ratio 265392 ratio / log m 3778.02
edit: I understand now: your function ( without the $H_n$) is multiplicative in the number theory sense. This means that multiplying by a new prime $p$ multiplies the "ratio" above by the function applied to $p$ itself, which is approximately $2.$ So the number I call "ratio" above is roughly doubling each time. Let
$$ f(n) =\frac{\sigma_0(n)\sigma_2(n)}{(\sigma(n))^2}. $$
biggest prime factor 2 ratio 1.11111 cumulative product 1.11111
biggest prime factor 3 ratio 1.25 cumulative product 1.38889
biggest prime factor 5 ratio 1.44444 cumulative product 2.00617
biggest prime factor 7 ratio 1.5625 cumulative product 3.13465
biggest prime factor 11 ratio 1.69444 cumulative product 5.31148
biggest prime factor 13 ratio 1.73469 cumulative product 9.2138
biggest prime factor 17 ratio 1.79012 cumulative product 16.4938
biggest prime factor 19 ratio 1.81 cumulative product 29.8538
biggest prime factor 23 ratio 1.84028 cumulative product 54.9393
biggest prime factor 29 ratio 1.87111 cumulative product 102.798
biggest prime factor 31 ratio 1.87891 cumulative product 193.147
biggest prime factor 37 ratio 1.89751 cumulative product 366.498
biggest prime factor 41 ratio 1.90703 cumulative product 698.922
biggest prime factor 43 ratio 1.91116 cumulative product 1335.75
biggest prime factor 47 ratio 1.9184 cumulative product 2562.51
biggest prime factor 53 ratio 1.9273 cumulative product 4938.71
biggest prime factor 59 ratio 1.93444 cumulative product 9553.67
biggest prime factor 61 ratio 1.93652 cumulative product 18500.9
biggest prime factor 67 ratio 1.94204 cumulative product 35929.5
biggest prime factor 71 ratio 1.94522 cumulative product 69890.7
biggest prime factor 73 ratio 1.94668 cumulative product 136055
biggest prime factor 79 ratio 1.95062 cumulative product 265392
biggest prime factor 83 ratio 1.95295 cumulative product 518296
biggest prime factor 89 ratio 1.95605 cumulative product 1.01381e+06
biggest prime factor 97 ratio 1.9596 cumulative product 1.98667e+06
Alright: from Merten's Theorem, if prime number $k$ is called $p,$ so that $p = p_k,$ the "cumulative product" is approximately a constant times
$$ \frac{2^k}{\log^2 p} $$