The Grothendieck-ring growth-dimension conjecture

Let $\mathcal C$ be a pretannakian category of moderate growth, and let $K_0(\mathcal C)$ be its Grothendieck ring. For an object $X$, let $\mathsf{gd}(X)$ denote its growth dimension, extended to a function $\mathsf{gd}:K_0(\mathcal C)\tomathbb R$.

Grothendieck-ring growth-dimension conjecture. The function

$$\mathsf{gd}:K_0(\mathcal C)\tomathbb R$$

is always a ring homomorphism.

Growth dimension would therefore behave multiplicatively on the Grothendieck ring of every moderate-growth pretannakian category. The statement is presented as a consequence of the higher Verlinde target conjecture and remains open in general.