Vertex-function formula for tangent characters under 3d-mirror symmetry

Let $(X,\mathfrak{C})$ and $(X',\mathfrak{C}')$ be pairs of symplectic varieties and chambers related by $3d$-mirror symmetry. For $p\in X^{\mathsf{T}}$, let $\textsf{b}(p)$ be the corresponding fixed point on $X'$. Suppose that

$$T_{\textsf{b}(p)}X'=N^+_{\textsf{b}(p)}\oplus N^-_{\textsf{b}(p)},$$

where $N^{\pm}_{\textsf{b}(p)}$ are the corresponding $\mathsf{T}'$-characters, and let $\Xi$ denote the multiplicative function defined from the $q$-analogue of the Gamma function. Let $\kappa^*$ denote substitution as in the source, and let $(N^-_{\textsf{b}(p)})^*$ be the $\mathsf{T}'$-module dual.

Vertex-function character formula. The vertex functions of $X$ with vanishing equivariant parameters are given by the Taylor series expansions of

$$\kappa^*{\bf V}_p(0_{\mathfrak{C}},{\bs z})=\Xi\left(q/\hbar',(N^-_{\textsf{b}(p)})^*\right).$$

This proposed formula expresses the tangent-space character of the $3d$-mirror directly in terms of the vertex functions of $X$, without requiring prior knowledge of $X'$. Since the $3d$-mirror is not known in most cases, the statement is presented as a proposal and remains open.