Pomerance's shifted-prime smoothness conjecture

For $X,Y\geq1$, let $P(n)$ be the greatest prime factor of $n$, and define

$$\Pi(X,Y)=\#\{p\leq X:P(p-1)\leq Y\}.$$

Let $\Psi(X,Y)$ denote the count of positive integers at most $X$ whose prime factors are all at most $Y$. Pomerance's conjecture. If $X,Y\to\infty$ with $X\geq Y$, then

$$\Pi(X,Y)\sim\frac{\Psi(X,Y)}{\log X}.$$

This asserts that shifted primes have the same relative smoothness frequency as generic integers. It is used as a conditional input in the paper and remains open in the stated generality.