Generalized $c4$-Li criterion for the class $aS^{\sharp \flat}(\sigma_0,\sigma_1)$

Let $F\in\mathcal{S}^{\sharp \flat}(\sigma_0,\sigma_1)$, where the positive real numbers $\lambda_j$ appearing in axiom (iii') determine

$$\mathcal{C}_F=\sum\limits_{j=1}^{r}\lambda_j.$$

Generalized $\tau$-Li criterion. Non-vanishing of $F$ in the half-plane $\operatorname{Re}(s)>\tau/2$ is equivalent to growth of $\lambda_F(n,\tau)$ as $\mathcal{C}_F\tau n\log n$ as $n\to\infty$.

This conjecture proposes an analogue, for the class $\mathcal{S}^{\sharp \flat}(\sigma_0,\sigma_1)$, of the criterion known for $\mathcal{S}^{\sharp\flat}$ when $\tau=1$, relating the zero-free half-plane to the asymptotic behavior of the generalized Li coefficients. Its status is unresolved in the supplied source.