Erdős problem 1138

Erdős Problem 1138. Let $x/2 < y < x$ and $C > 1$. If $d = \max_{p_n < x}(p_{n+1} - p_n)$, where $p_n$ denotes the $n$-th prime, then is it true that $$\pi(y + Cd) - \pi(y) \sim \frac{Cd}{\log y}$$?