Uniformity conjecture for large-prime square divisors of discriminants

Let $V$ be the representation in the paper, let $\underline{V}(\mathbb{Z})^{irr}$ denote its irreducible integral elements, let $\Delta$ be the discriminant, and let $N(\,3, X)$ be the counting function used in the paper. For a prime $p$, define

$$\mathcal{W}_p(V)=\{v\in\underline{V}(\mathbb{Z})^{irr}:p^2\mid\Delta(v)\}.$$

Uniformity conjecture. For every $M>0$,

$$\lim_{X\to+\infty}\frac{N(\bigcup_{p>M}\mathcal{W}_p(V),X)}{X^{\dim V}}=O\left(\frac{1}{\log M}\right),$$

where the implied constant is independent of $M$. This is the direct analogue of the uniformity estimate used in the corresponding 2-Selmer counting problem, and would provide the lower-bound input needed when infinitely many congruence conditions are imposed. Its resolution is not stated in the source.