Folklore generic Newton polygon conjecture for triangular exponential sums

Let $\Delta$ be a triangle with vertices at $(0,0)$, $\mathbf P_1=(a_1,b_1)$, and $\mathbf P_2=(a_2,b_2)$. Assume the hypotheses on the prime $p$ from the main theorem. Let $\mathrm{NP}(f,\chi)_C$ denote the corresponding Newton polygon for a polynomial $f(x_1,x_2)$ whose Newton polytope is the convex hull $\Delta$, and let $\mathrm{IHP}(\Delta)$ be the associated infinite Hodge polygon. Folklore conjecture. In the moduli space of all such polynomials, there exists an open dense subspace over which, for every finite character $\chi$, every integer $k\geq 1$, and every integer $i_k$ satisfying $0\leq i_k\leq \mathbb x_k'-\mathbb x_k$, the Newton polygon agrees with the infinite Hodge polygon at

$$x=\mathbb x_k+i_k.$$

The claim concerns generic polynomials with triangular Newton polytope and would provide the coincidence needed to apply the paper's main theorem. The supplied context describes it as a folklore conjecture and gives no evidence that it has been resolved.