Type II distribution conjecture

Let $0<\varpi<1/4$, and let $\delta>0$ be a sufficiently small fixed quantity depending on $\vartheta$. Let $x$ tend to infinity, let $P$ be a square-free number all of whose prime factors are at most $x^\delta$, and let $N,M$ satisfy

$$x^{1/2-2\varpi}\ll N\ll M\ll x^{1/2+2\varpi},\qquad NM\asymp x.$$

Let $\alpha,\beta\colon\mathbb{N}\to\mathbb{R}$ be supported on $[M,2M]$ and $[N,2N]$, respectively, with $|\alpha(n)|\ll1$, and set $\beta=1_{[N,2N]}$. Type II conjecture. For any fixed $A>0$,

$$\sup_{1\leq a\leq x:(a,P)=1}\sum_{r\ll x^{1/2+2\varpi}:r\mid P}\left|\Delta(\alpha\star\beta;a\ (r))\right|\ll x\log^{-A}x.$$

This conjecture is introduced as an alternative distributional hypothesis: partial progress toward it would yield partial progress on Vinogradov's least quadratic nonresidue conjecture. The supplied material does not give a resolution.