Volume entropy monotonicity conjecture for the G-alpha family

For $-1\leq\alpha\leq 1$, let $G_\alpha$ be the homogeneous Riemannian manifold in the interpolating family, and let $h(G_\alpha)$ denote its volume entropy, defined by

$$h(G_\alpha):=\lim_{R\rightarrow\infty}\frac{\log(\operatorname{Vol} B(R))}{R},$$

where $B(R)$ is a geodesic ball of radius $R$. Volume entropy monotonicity conjecture. The function $h(G_\alpha)$ is monotonically decreasing in $\alpha$ for $\alpha\in[-1,1]$. The endpoint values given in the text are $h(G_{-1})=2$ and $h(G_1)=1$; the conjecture seeks quantitative control of entropy along the interpolation from hyperbolic space to Sol.