Brauer–Siegel conjecture for constant algebraic tori

Let $K$ be the function field in the preceding setting, and let $T=T_0\times_{\mathbb F_q}K$, where $T_0$ is a fixed $\mathbb F_q$-torus. Put $d=\dim T$, $N_m(T_0)=|T_0(\mathbb F_{q^m})|$, and let $\mathcal D_T$ be the quasi-discriminant of $T$. Let $h(T)$ denote the class number of $T$, and let $\beta_m$ have the meaning introduced in the preceding sections. Constant-torus Brauer–Siegel conjecture.

$$\lim_{g\to\infty}\frac{1}{g}\log h(T)=\lim_{g\to\infty}\frac{1}{g}\log\sqrt{\mathcal D_T}-\sum_{m=1}^{\infty}\beta_m\log_q\frac{N_m(T_0)}{q^{md}}.$$

This is proposed as an analogue of the Brauer–Siegel theorem for algebraic tori, motivated by known analogues of class numbers, regulators, and analytic class number formulas; its status is open.