Conjecture on the radius of convergence for generalized Lambert W inverse series

Let $f(z)=z\prod_{j=1}^m(z-t_j)^{p_j}e^z$, where the $t_j$ and $p_j$ are nonzero complex numbers, and let $f^{-1}(z)$ be its inverse branch at $0$, whose Taylor series is given by

$$f^{-1}(z)=\sum_{n=1}^{\infty}\frac{(-n)^{n-1}}{n!}(-t_1)^{-np_1}\cdots(-t_m)^{-np_m}F_nz^n.$$

Radius-of-convergence conjecture. There exist integers $l_1,\ldots,l_m$ such that the radius of convergence of this series is

$$R=\left|\exp\left(2\pi i\sum_jl_j\phi_j-1\right)\left(1-\sum_j\phi_j\right)^{1-\sum_j\phi_j}\prod_j\left(\frac{t_j}{p_j+\phi_j}\right)^{p_j+\phi_j}p_j^{p_j}\phi_j^{\phi_j}\right|,$$

where $\phi$ is one of the solutions of

$$\left(1-\sum_j\lambda_j\right)(p_i+\lambda_i)+\lambda_it_i=0.$$

The conjecture identifies the convergence radius through the critical points governing the inverse function's analytic continuation. The source gives no resolution, so the status of this formula remains open.