Vemulapalli's scrollar-invariant polytope conjecture for primitive covers

Let $k\geq 3$, let $g\geq 0$, and consider the polytope

$$\mathcal P_k:=\left\{\mathbf{x}\in\mathbb R^{k-1}\mid \sum_{i\leq k-1}x_i=1,\ 0\leq x_1\leq\dots\leq x_{k-1},\ x_{i+j}\leq x_i+x_j\right\}.$$

A scrollar-invariant tuple is a tuple of positive integers $\mathbf e=(e_1,\dots,e_{k-1})$ with $e_1+\dots+e_{k-1}=g+k-1$. Vemulapalli's conjecture. Such a tuple arises as the scrollar invariants of a primitive degree-$k$ cover $\pi:C\rightarrow\mathbb P^1_{\mathbb k}$ from a genus-$g$ curve if and only if

$$\left(\frac{e_1}{g+k-1},\dots,\frac{e_1}{g+k-1}\right)\in\mathcal P_d.$$

The conjecture gives a polyhedral characterization of the scrollar invariants that occur for primitive covers of the projective line. The supplied text attributes it to Vemulapalli and states no evidence of a resolution.