Conjecture on unequal digits in the base-$b$ expansion of $Z_b(b^k)$

Let $b\geq 2$ and $k\geq 1$. Write $Z_b(b^k)$ for the number of trailing zeroes of $b^k!$, and let $\vartheta(b)$ be the limiting constant whose first $k$ base-$b$ digits are compared with those of $Z_b(b^k)$. The number of unequal digits between $Z_b(b^k)$ and the first $k$ digits of the base-$b$ expansion of $\vartheta(b)$ is smaller than or equal to the number of digits of $k$ plus $1$.

Digit-discrepancy conjecture. The number of unequal digits between $Z_b(b^k)$ and the first $k$ digits of the base-$b$ expansion of $\vartheta(b)$ is smaller than or equal to the number of digits of $k$ plus $1$.

The preceding bound was computationally observed to be strict in the authors' tests, but no proof was known. The conjecture concerns the accuracy of the base-$b$ expansion of the asymptotic approximation $\vartheta(b)b^k$ to the number of trailing zeroes of $b^k!$.