The semi-arc k-visibility graph edge-maximization conjecture

A semi-arc $k$-visibility graph is a graph represented by semi-arcs, with visibility allowing at most $k$ obstructing semi-arcs; let $n$ be the number of vertices. Semi-arc $k$-visibility graph edge-maximization conjecture. The maximum number of edges in a semi-arc $k$-visibility graph on $n$ vertices is

$$(k+1)\left(2n-\frac{3k+6}{2}\right)$$

for $n \geq 3k+3$. This conjecture would determine the extremal edge count beyond the open range identified in the paper; the stated bound is conjectured to be tight.