Binary-ternary collisions and the last significant digit of $n!$ in base 12

The third-named author recently proved [Israel J. of Math. 258 (2023), 475–502] that there are infinitely many $\mathit{\text{collisions}}$ of the base-2 and base-3 sum-of-digits functions. In other words, the equation $$s_2(n)=s_3(n)$$ admits infinitely many solutions in natural numbers. We refine this result and prove that every integer $a$ in $\{1,2,\dots ,11\}$ appears as the last nonzero digit of $n!$ in base $12$ infinitely often.