Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to this MO post.

For integer $n\ge0$, let $s(n)$ denote the sum of the digits in the decimal representation of $n$.

Is it true that for any integer $a,b>0$, the ratio of which is not a power of $10$, the set of all those $n\ge 0$ with $s(an)=s(bn)$ has zero density?