The Kikuta–Takahashi categorical Gromov–Yomdin conjecture

Let $X$ be a smooth projective variety over $\mathbb{C}$ and let $D^b(X)$ denote its bounded derived category of coherent sheaves. Let $\mathcal{N}(X)$ be the numerical Grothendieck group of $X$, and let $\Phi: D^b(X) \to D^b(X)$ be an exact autoequivalence inducing an automorphism $[\Phi]$ on $\mathcal{N}(X) \otimes_\mathbb{Z} \mathbb{C}$. Write $\rho([\Phi])$ for the spectral radius of this induced automorphism. Kikuta–Takahashi conjecture. One has

$$h_0(\Phi)=\log\rho([\Phi]).$$

This is the categorical analogue of the Gromov–Yomdin theorem, relating categorical entropy to the spectral radius of the induced action on numerical K-theory. The source presents it as an application of the paper's results; its resolution is not established by the supplied text.