Matrix-element conjecture for the crystal-limit vertex operator

Let $\widetilde\Phi(z):\mathcal F_{\vec u}\to\mathcal F_{\vec v}$ be the crystal-limit vertex operator, with $\vec u=(u_1,u_2)$ and $\vec v=(v_1,v_2)$, and let $|\widetilde K_{\vec\mu}\rangle$ and $\langle\widetilde K_{\vec\lambda}|$ be integral-form generalized Hall–Littlewood vectors. Let $\widetilde N_{\lambda,\mu}(Q)$ denote the corresponding Nekrasov factor. Crystal-limit vertex-operator conjecture.

$$\begin{aligned} \langle\widetilde K_{\vec\lambda}|\widetilde\Phi(z)|\widetilde K_{\vec\mu}\rangle &=(-1)^{|\vec\lambda|+|\vec\mu|}(u_1u_2v_1v_2z)^{|\vec\lambda|-|\vec\mu|}u_1^{2|\mu^{(1)}|}u_2^{2|\mu^{(2)}|}(u_1u_2)^{|\vec\mu|}\\\\ &\quad\times t^{-2(n(\mu^{(1)})+n(\mu^{(2)}))}\prod_{i,j=1}^{2}\widetilde N_{\lambda^{(i)},\mu^{(j)}}(v_i/u_j). \end{aligned}$$

This is the crystal-limit counterpart of the generic-$q$ intertwiner formula; the equality is marked conjectural and no resolution is supplied.