Flexibility above the Besson–Courtois–Gallot threshold

Let $M^n$ be a closed manifold admitting a negatively curved locally symmetric metric $g_0$. Define

$$C(M):=h_{\mathrm{top}}(\varphi_{g_0})\,\mathrm{vol}_{d\lambda}(D_{g_0}^*M)^{1/n}.$$

Flexibility above the Besson–Courtois–Gallot threshold. For every

$$c>C(M),$$

there exists a negatively curved Riemannian metric $g$ on $M$ such that

$$\mathrm{vol}_{d\lambda}(D_g^*M)=1$$

and

$$\hbar_{\rm per}(D_g^*M,\lambda_{\mathrm{can}})=\hbar_{\rm bar}(D_g^*M,\lambda_{\mathrm{can}})=c.$$

This is the higher-dimensional entropy-flexibility problem above the Besson–Courtois–Gallot threshold. The supplied text does not establish the statement or indicate whether it has been resolved.