The affine surface hypothesis for polynomials with irreducible leading form

Let $\delta\geq 2$. For a polynomial $g\in\mathbb{Z}[T_1,\ldots,T_\nu]$, let $\mathfrak{h}(g)$ denote its homogeneous part of maximal degree, and let $\mathcal{J}_\delta$ be the set of degree-$\delta$ polynomials $f\in\mathbb{Z}[T_1,T_2,T_3]$ for which $\mathfrak{h}(f)$ is irreducible. Let $\mathsf{ASH}[\alpha,\mathcal{P}]$ denote the assertion that, for every $\varepsilon>0$, there is a constant depending only on $\delta$ and $\varepsilon$ such that $M(f;B)=O_{\delta,\varepsilon}(B^{\alpha+\varepsilon})$ for every $f\in\mathcal{P}$. Affine surface hypothesis. For every $\delta\geq 2$, $\mathsf{ASH}[1,\mathcal{J}_\delta]$ holds. The paper has already proved the hypothesis with exponent $1+1/\delta$ for all irreducible degree-$\delta$ polynomials, and proves stronger exponents for the restricted classes $\mathcal{J}_\delta$ in several degree ranges. The conjecture asserts the optimal exponent $1$ for polynomials whose highest-degree homogeneous part is irreducible.