Volume-entropy generalization of the Fang–Zhang–Zhang conjecture

Let $X$ be a closed oriented smooth Riemannian $4$-manifold, and let $\mu(X)$ denote its volume entropy. Assume that

$$\mu(X)\ne 0,\qquad \bar{\lambda}(X)<0,$$

and that $X$ admits a quasi-non-singular solution to the normalized Ricci flow. Volume-entropy generalization of the Fang–Zhang–Zhang conjecture. Then

$$2e(X)-3|\operatorname{sign}(X)|\geq \frac{1}{54\pi^2}\mu(X)^4.$$

This conjecture extends the FZZ inequality from simplicial volume to volume entropy and includes the FZZ conjecture as a special case. The source presents it as a proposed generalization; its resolution is not established there.