Erdős–Pomerance–Sarkőzy conjecture on the order of equal consecutive divisor values

Let $\tau$ be the divisor function and define

$$S(x):=\left|\{n\leq x:\tau(n)=\tau(n+1)\}\right|.$$

Erdős–Pomerance–Sarkőzy conjecture.

$$S(x)\asymp x(\log_2 x)^{-\frac{1}{2}}.$$

This refines the proved infinitude result for integers with equal divisor counts at consecutive arguments by predicting the precise order of magnitude of their counting function. The source gives no resolution of this stronger conjecture.