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    $\begingroup$ As I understood,, if the first number in the sequence of consecutive numbers ends in 96 and the rest of the number is the same for all numbers in the sequence then the sequence consists of at most four numbers. $\endgroup$
    – Alex Ravsky
    Commented Mar 15 at 19:03
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    $\begingroup$ @AlexRavsky Frankly, I hadn't given any thought to the case where the 'start' of the numbers changes. In this case the next number will end in $d000\ldots00$ where $d$ is one greater than one of the original digits, which is either $1$ or even. So $d=2$ or $d$ is odd, and by the same observations $d\neq5$ and $d\neq7$. So $d\in\{1,2,3,9\}$. Not sure if we can say more. $\endgroup$
    – user88375
    Commented Mar 15 at 20:37