For every base $b$, is the concatenation sequence of $b$ a strong divisibility sequence?

Question

Let $b$ be an integer greater than or equal to $2$, and consider base $b$. The concatenation sequence associated with $b$ is the sequence of positive integers $s_n$ such that the $n$-th number in the sequence, is the number $n...n$. That is, you write $n$ in base $b$, concatenate it with itself $n$ times, and then return the resulting number in base $b$. For example, in our regular base ten, $s_1=1$, $s_2=22$, $s_3=333$, etc. My question is, is the concatenation sequence of every base, a strong divisibility sequence? The definition of a strong divisibility sequence is a sequence such that $m$ divides $n$ if and only if $s_m$ divides $s_n$. Given some numerical evidence, I conjecture that it is, but of course that is not a proof. Can anyone prove it?