Strict-corner LLT e-positivity difference conjecture

Let $\Gamma$ be a circular unit arc digraph, and let $H$ be a graph obtained from $\Gamma$ by marking $k$ corner edges strict. Strict-corner difference conjecture. Then

$$\mathrm{G}_\Gamma(\mathbf{x};q+1)-q^k\mathrm{G}_H(\mathbf{x};q+1)$$

is $\mathrm{e}$-positive. The conjecture is indicated by computer experiments and remains open.