Wada's decomposition conjecture for Cartan matrix eigenvalues and elementary divisors

Let $B$ be a block of a finite group $G$ over an algebraically closed field of characteristic $p>0$, let $C\in\mathbb{Z}^{l\times l}$ be its Cartan matrix, and let $D$ be a defect group of $B$. Write $ED$ for the multiset of elementary divisors of $C$, $EV$ for the multiset of eigenvalues of $C$, and let $\rho(C)$ denote the spectral radius of $C$. Wada's decomposition conjecture. There exist partitions of multisets

$$EV=E_1\sqcup\ldots\sqcup E_n,\qquad ED=F_1\sqcup\ldots\sqcup F_n$$

such that, for $i=1,\ldots,n$, $|E_i|=|F_i|$, $\prod_{\lambda\in E_i}\lambda=\prod_{\lambda\in F_i}\lambda$, and

$$\prod_{\lambda\in E_i}(X-\lambda)\in\mathbb{Z}[X]$$

is irreducible; moreover, $\rho(C)\in E_1$ and $|D|\in F_1$. The conjecture is presented as a strengthening of implications concerning the rationality and maximality of eigenvalues; the paper does not report a resolution, so it remains open.