Inspired by this question, is there some conjecture stating that $$ \limsup_{n \to \infty} \frac{d_j(2^n)}{dc(2^n)} = \frac{1}{10} $$ where $d_j(m)$ counts the number of $j$s in the digits of $m$, and $dc(m)$ is just the number of digits in $m$?

Or is this false for obvious reasons? If so, what is this limit? Is the limit same as the $\liminf$?

This can of course be stated in a more general form, where we use another number than 2, and another basis than 10. There are of course some trivial combinations where the corresponding statement is clearly false, for example, if we count binary digits in the above problem (base 2).

This is probably very hard to solve, since it is closely related to normal numbers.