Vemulapalli–Vishwanath conjecture on primitive cover scrollar invariants

Let $d$ be a degree and let $\vec{e}=(e_1,\ldots,e_{d-1})$ be a $(d-1)$-tuple of scrollar invariants. A cover is primitive if it does not factor through a nontrivial proper subcover. Vemulapalli–Vishwanath conjecture. A $(d-1)$-tuple $\vec{e}$ arises as the scrollar invariants of a smooth primitive degree $d$ cover if and only if

$$e_{i+j}\leq e_i+e_j$$

for all $i,j$. The conjecture proposes that the known inequalities are also sufficient for realizability by smooth primitive covers; the paper invokes it as an open question and proves the relevant realizability result for smooth sextic covers without establishing the primitive case.