The W-algebra conformal-block conjecture for multi-leg PT vertices

Let $W_{q_{1,2,3,4}}(\widehat{A}_{0})$ be the W-algebra associated with the quantum parameters $q_{1,2,3,4}$, and consider conformal blocks of multi-screened vertex operators. The associated higher-rank multi-leg Pandharipande–Thomas (PT) vertices impose boundary conditions in multiple directions of $\mathbb{C}^3$ or $\mathbb{C}^4$. Multi-leg PT vertex conjecture. The conformal block of multi-screened vertex operators of the W-algebra $W_{q_{1,2,3,4}}(\widehat{A}_{0})$ is the higher-rank multi-leg PT vertex of $\mathbb{C}^3$ and $\mathbb{C}^4$. This identifies the algebraic conformal-block construction with the gauge-origami partition function and the PT invariants for the corresponding higher-dimensional geometries; the authors derive contour-integral formulas and check their pole structures against PT counts for various boundary conditions, but present the general identification as a conjecture.