Conjectural tail estimates for discriminant-divisible elements

Let $n$ be the degree parameter, let $A$ and $b$ specify the space $V_{A,b}$, and let $\Delta(v)$ denote the discriminant of $v\in V_{A,b}(\mathbb{Z})$. For each prime $p$, define

$$\mathcal{W}_{A,b,p}=\{v\in V_{A,b}(\mathbb{Z}):p^2\mid\Delta(v)\}.$$

For $X$ and $M$, let $N(S,X)$ count the elements of $S$ of height at most $X$.

Conjectural tail estimates. For every $\epsilon>0$,

$$N\left(\bigcup_{p\geq M}\mathcal{W}_{A,b,p},X\right)=O\left(\frac{X^{\frac{n(n+1)}{2}-1}}{M^{1-\epsilon}}\right)+o\left(X^{\frac{n(n+1)}{2}-1}\right),$$

where the implied constant is independent of $X$ and $M$.

These estimates control the contribution from elements whose discriminant is divisible by the square of a large prime, and are used to sieve local specifications into acceptable sets. They are known for $n=4$ in forthcoming work with Arul Shankar and are likely to hold for $n\geq 6$; suitable versions of the $abc$ conjecture imply them by work of Granville.