Erdős's valuation growth conjecture for central binomial coefficients

Let $p$ be an odd prime. For an integer $a$, define $\epsilon_p(a)$ by

$$p^{\epsilon_p(a)}\mid\mid a.$$

Erdős's valuation growth conjecture. The $p$-adic valuation of the central binomial coefficient satisfies

$$\epsilon_p\left(\binom{2^{k+1}}{2^k}\right)=\frac{\log(2)}{2\log(p)}\,k+O(\sqrt{k}).$$

This is motivated by a heuristic of uniformly distributed digits in the base-$p$ representations of powers of $2$ and by computer experiments for primes $p<200$. The source gives no proof or resolution.