The higher-level Whittaker-vector expansion conjecture

Let $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_m)$ and $\boldsymbol{\beta}=(\beta_1,\ldots,\beta_m)$ be parameter tuples, and let $\left|G;\Lambda,\boldsymbol{\alpha}\right>$ and $\left<G;\Lambda,\boldsymbol{\beta}\right|$ be the normalized higher-level Whittaker vector and dual state in the completed Fock space, defined by the displayed Ding–Iohara mode conditions. Write $v=\sqrt{q/t}$, $\Lambda_{\boldsymbol{\alpha}}=\Lambda\prod_i vu_i/\alpha_i$, and $\Lambda_{\boldsymbol{\beta}}=\Lambda\prod_i vu_i/\beta_i$. Higher-level Whittaker expansion conjecture. For generic parameters, both states exist uniquely, and their coefficients in the generalized Macdonald basis are given by the explicit product formulas stated in the source: the ket coefficients $C_{\boldsymbol\lambda}(\Lambda,\mathbf{u},\boldsymbol{\alpha};q,t)$ equal the displayed arm-, leg-, and Nekrasov-factor product, while the dual coefficients satisfy the displayed reversed-tuple and parameter-inversion identities and their two equivalent product formulas. This conjecture gives explicit higher-level Whittaker expansions and is presented as the paper's main conjecture; no resolution is supplied.