Weak Granville conjecture for smooth shifted primes in a quadratic setting

For $X,Y\to\infty$ with $X\geq Y$, define

$$U=\frac{\log X}{\log Y}$$

and let

$$\Pi^{*}(X,Y)=\#\{p\leq X:P(p-(\Delta/p))\leq Y\},$$

where $(\Delta/p)$ is the Legendre symbol and $\Psi(X,Y)$ counts the $Y$-smooth integers at most $X$. Weak Granville conjecture. If

$$U=\exp((1+o(1))\sqrt{\log X}),$$

then

$$\Pi^{*}(X,Y)\gg\frac{\Psi(X,Y)}{\log X}.$$

A fixed-$U$ instance follows under the weak Elliott–Halberstam conjecture, while the larger range asserted here is used for the paper's conditional lower bound and is not proved.