Oscillation criterion for zeros and generalized Li coefficients

Let $F\in\mathcal{S}^{\sharp \flat }(\sigma_0,\sigma_1)$, let $\tau$ be the parameter in the generalized Li coefficients $\lambda_F(n,\tau)$, and consider the half-plane $\operatorname{Re}(s)>\tau/2$. The generalized Li oscillation conjecture. The function $F$ possesses non-trivial zeros in this half-plane if and only if the coefficients $\lambda_F(n,\tau)$ oscillate with exponentially growing amplitude as $n\to\infty$. This is the companion conjecture to the asymptotic criterion above and is based on the paper's numerical evidence; no resolution is supplied.