Ruzsa's short-interval divisor conjecture

For integers $N$ and real $a\leqslant b$, let $\tau(N;a,b)$ denote the number of divisors of $N$ in $[a,b]$. Ruzsa's conjecture. For every $\varepsilon>0$ there is $C(\varepsilon)>0$ such that every positive integer $N$ satisfies

$$\tau(N;N^{1/2},N^{1/2}+N^{1/2-\varepsilon})\leqslant C(\varepsilon).$$

The conjecture is known for $\varepsilon>1/4$, while the full assertion is open. The paper notes its close similarity to the paper's short-interval divisor conjecture.