Existence and polynomial dependence of ramified irregular vertex operators

Let $r$ be a positive integer and let $M^{[r]}_{\Lambda}$ be an irregular Verma module with $\Lambda_{2r-1}\neq 0$ and $\Lambda_{2r}=0$. A ramified irregular vertex operator has an expansion of the form

$$\Phi^{\Delta}_{\Lambda',\Lambda}(z)|\Lambda\rangle=z^{\alpha}\exp\left(\sum_{i=1}^{2r-1}\frac{\beta_i}{z^{i/2}}\right)\sum_{m=0}^{\infty}v_mz^{m/2},$$

with $v_0=|\Lambda'\rangle$. Ramified vertex-operator existence conjecture. The operator exists, with $\Lambda'=\Lambda$ and

$$v_m=\sum_{|\mu|\leq m}c_{\mu}^{(m)}L_{-\mu}|\Lambda\rangle,$$

where each $c_{\mu}^{(m)}$ is a polynomial in $\alpha$, $\beta_1,\ldots,\beta_{2r-1}$, $\Lambda_r,\ldots,\Lambda_{2r-1}$, $\Lambda_{2r-1}^{-1}$, and $c_{\emptyset}^{(k)}$ for $k\leq m-1$. The source gives this as a conjecture and does not report a resolution.