The conjecture for categories associated with the subregular nilpotent in $G_2$

Let $l\geq 7$ be an integer not divisible by $3$, and let $q$ be the parameter defining the braided finite tensor category $\mathcal{C}(G_2,G_2(a_1),l,q)$. Write $T(\omega_1)$ and $T(\omega_2)$ for the tilting modules of highest weights $\omega_1$ and $\omega_2$, and let $[r]_l$ denote the quantum integer. The conjecture for $\mathcal{C}(G_2,G_2(a_1),l,q)$.

(1) The Frobenius–Perron dimensions satisfy

$$\operatorname{FPdim}(\mathcal{C}(G_2,G_2(a_1),l,q))=6\frac{l^4}{(2\sin(\pi/l))^8(2\sin(2\pi/l))^2},$$

and

$$\operatorname{FPdim}(T(\omega_1))=2[3]_l+1,\qquad \operatorname{FPdim}(T(\omega_2))=[5]_l+3[3]_l.$$

(2) The category $\mathcal{C}(G_2,G_2(a_1),l,q)$ has the stable Chevalley property.

(3) Its Müger center is equivalent to $\operatorname{Rep}(S_3)$. The projective covers of the simple objects from $\operatorname{Rep}(S_3)$ lie in the principal block, and the highest weights of the corresponding tilting modules lie in the alcoves $C_0$, $C_4$, and $C_7$. The formulas have been checked for $l=7,11$ and are compatible with the separately described $l=5$ semisimplification; the general case remains conjectural.