LLT e-positivity conjecture for circular Dyck diagrams

Let ${\boldsymbol\nu} = (\mathbf{a},\mathbf{s})$ be a circular Dyck diagram with some strict corner edges. Circular LLT e-positivity conjecture. Then $\mathrm{G}_{\boldsymbol\nu}(\mathbf{x};q+1)$ is $\mathrm{e}$-positive with unimodal coefficients. This is the paper's main conjecture, supported by several results, but remains open.