Shellability conjecture for k-cut complexes of hexagonal line tilings

Let $k$ be one of $4$, $5$, or $6$, and let $H$ be a hexagonal line tiling. Consider the $k$-cut complex of $H$.

Hexagonal line tiling shellability conjecture. For $k=4,5,$ and $6$, the $k$-cut complex of the hexagonal line tiling is shellable.

The conjecture is motivated by computational experiments using SageMath, which suggest that the behavior proved in the paper for $3$-cut complexes persists for these higher values of $k$. No resolution is given in the supplied source.