The weak digit-one density conjecture for initial base-q digits

Let $p,q$ be two distinct prime numbers. For each $k$ and each admissible integer $m$, let $d_{k,m}$ denote the number of digit ones among the first $m$ digits of the $q$-ary expansion of $p^k$. Weak digit-one density conjecture. There is a non-decreasing integer sequence $m(k)$ tending to infinity, with $m(k)\leq k\log p/\log q$, such that

$$\lim_{k\to\infty}\frac{d_{k,m(k)}}{m(k)}=\frac{1}{q}.$$

The paper describes this as weaker than the preceding digit-one density conjecture and notes that it would imply the Erdős ternary problem.