Sharp entropy-at-infinity upper bound for uniformly continuous potentials

Let $(M,g)$ be a pinched negatively curved manifold. Let $(\mu_n)_n$ be a sequence of invariant probability measures converging vaguely to $\mu$, and let $F\in C_b(T^1M)$ be a uniformly continuous potential. Sharp entropy-at-infinity conjecture. Then

$$\limsup_{n\to\infty} \bigg(h_{\mu_n}(g)+\int Fd\mu_n\bigg)\leqslant |\mu|\bigg(h_{\mu/|\mu|}(g)+\int Fd\mu/|\mu|\bigg)+(1-|\mu|)\delta^F_{\Gamma,\infty}.$$

This conjecture asserts that the pressure contribution from mass escaping to infinity is governed by the entropy at infinity of the potential, improving the general bound involving $\delta_\infty+c(F)$. Its resolution is not supplied in the source context.