Vemulapalli conjectural characterization for primitive coverings

Prove or disprove that a non-decreasing sequence $(e_1,\dotsc,e_{d-1})$ is the scrollar-invariant sequence of a primitive degree-$d$ covering of genus $g$ if and only if every $e_i\ge 1$, the sum satisfies $\sum_i e_i=d+g-1$, and $e_i+e_j\ge e_{i+j}$ for all admissible indices $i,j$.

Background

The paper describes a conjectural criterion proposed by Vemulapalli for the realizability of scrollar invariants of primitive covers. The criterion consists of positivity, the degree-genus sum constraint, and a subadditivity condition.

The cited work establishes the necessity of these conditions and realizes a positive proportion of sequences in a suitably scaled polytope, but does not establish the converse in full. The conjecture concerns primitive coverings; the paper also notes that the subadditivity condition can fail for non-primitive coverings.