Cartan group conjecture for graded invariant factors

Let $ell\geq1$ and $d\geq0$, and let $X_{ell,d}$ be the matrix arising in the computation of Cartan invariants. Write the prime decomposition as $ell=\prod_i p_i^{r_i}$ and define the graded invariant factors by

$$\vartheta_{ell}(\lambda)=\prod_i\vartheta_{p_i^{r_i}}(\lambda),\qquad \lambda\in\operatorname{Par}(d).$$

Let $\operatorname{Cart}(ell,d)$ be the finite group associated with $X_{ell,d}$. Cartan group conjecture. For any $ell$, the finite group $\operatorname{Cart}(ell,d)$ is a direct product of cyclic groups with orders given by the graded invariant factors of X_{ell,d. This is an explicit group-theoretic formulation of the proposed invariant-factor description; the supplied context does not state a resolution of this claim.