The factorized vertex-operator matrix-element conjecture

Let $\mathbf{u}=(u_1,\ldots,u_m)$ and $\mathbf{v}=(v_1,\ldots,v_m)$, and let $\Phi(w)=\Phi_{\mathbf{u}}^{\mathbf{v}}(w):\mathcal{F}_{\mathbf{u}}\to\mathcal{F}_{\mathbf{v}}$ be the vertex operator defined by its vacuum normalization and intertwining relation with the Ding–Iohara algebra. Let $\left|K_{\boldsymbol \lambda}\right>$ and $\left<K_{\boldsymbol \lambda}\right|$ denote the integral forms. Factorized vertex-operator conjecture. (1) The operator $\Phi(w)$ exists uniquely. (2) Its matrix elements satisfy

$$\left<K_{\boldsymbol \lambda}\right|\Phi(w)\left|K_{\boldsymbol \mu}\right> = \left((-1)^m(t/q)^m e_m(\mathbf{u})e_m(\mathbf{v})w\right)^{|{\boldsymbol \lambda}|}\left((t/q)e_m(\mathbf{v})w\right)^{-|{\boldsymbol \mu}|}$$ $$\times\prod_{k=1}^m v_k^{-(m-1)|\lambda^{(k)}|}u_k^{|\mu^{(k)}|}q^{-(m-1)n(\lambda^{(k)'})+n(\mu^{(k)'})}t^{(m-1)n(\lambda^{(k)})-n(\mu^{(k)})} \prod_{i,j=1}^mN_{\lambda^{(i)},\mu^{(j)}}(qv_i/(tu_j)).$$

This conjecture proposes both existence and uniqueness of the higher-level vertex operator and a Nekrasov-factorized formula for its integral-basis matrix elements; no resolution is given in the source.