General version of James' conjecture for exceptional Hecke algebras

Let $W$ be a finite Weyl group, let $\mathcal H_k(W,\xi)$ be its Hecke algebra over a field $k$ with parameter $\xi$, and let $e<\infty$ satisfy the standing assumptions $\operatorname{char}(k)=\ell>0$ and that $\ell$ is a good prime for $W$. Assume that $e\ell$ does not divide any degree of $W$. Let $\Lambda$ index the Specht modules $W_\xi^\lambda$ and let $L_\xi^\mu$ denote the irreducible modules indexed by $\mu\in\Lambda_\xi^\circ=\Lambda_{\zeta_e}^\circ$. General version of James' conjecture. The decomposition matrix $D_\xi$ depends only on $e$; equivalently, the adjustment matrix $A_\xi^e$ is the identity, so that

$$(W_\xi^\lambda:L_\xi^\mu)=(W_{\zeta_e}^\lambda:L_{\zeta_e}^\mu)$$

for all $\lambda\in\Lambda$ and $\mu\in\Lambda_\xi^\circ=\Lambda_{\zeta_e}^\circ$. The conjecture is a generalization of James' type-$A$ prediction to Hecke algebras of finite Weyl groups; the supplied text records the necessary indexing-set equality but gives no resolution of the conjecture itself.