The lower-bound conjecture for polynomial volume growth

Let $X$ be a normal projective variety of dimension $d\ge 2$ over $\mathbf{k}$, and let $f$ be an automorphism of $X$ such that $\deg_1(f^n) \asymp n^{k}$ as $n\to\infty$. Lower-bound conjecture.

$$\operatorname{plov}(f) \ge d+\frac{k(k+2)}{4}.$$

This conjecture gives a lower bound for polynomial volume growth in terms of the degree-growth exponent. The supplied text states that it remains open for $d\ge 5$ and $k=2d-4$, where it predicts $\operatorname{plov}(f)\ge (d-1)^2+1$.