Zero-dimensional quiver-variety vertex-function factorization conjecture

Let $Q$ be a quiver without loops, let $\mathcal{M}=\mathcal{M}_{Q,\theta}(\mathsf{v},\mathsf{w})$ be a zero-dimensional quiver variety with $\theta=(1,1,\ldots,1)$, and let $\mu=\lambda-\sum_{i\in Q_0}\mathsf{v}_i\alpha_i$, where $\lambda=\sum_{i\in Q_0}\mathsf{w}_i\varpi_i$. Write $\Phi(x)=\prod_{i=0}^{\infty}(1-xq^i)$ and let $e^{\alpha_i}=z_i(q/\hbar)^{b_i}$. Zero-dimensional vertex-function factorization conjecture. The vertex function of $\mathcal{M}$ factorizes

$$\mathsf{V}_{\mathcal{M}}(z)=\prod_{\alpha\in\Phi^{+,\mathrm{re}}_\mu}\prod_{i=1}^{-(\alpha,\mu)}\frac{\Phi\left(\hbar\left(\frac{\hbar}{q}\right)^{i-1}e^{\alpha}\right)}{\Phi\left(\left(\frac{\hbar}{q}\right)^{i-1}e^{\alpha}\right)}.$$

This is a conjectural explicit factorization for vertex functions of zero-dimensional quiver varieties; the source notes proofs in several cases, including type $A$ with one framing, type $D$ with a single minuscule framing, and cotangent bundles of Grassmannians, while the general case remains open.