Lapid–Mínguez conjecture on Jordan–Hölder constituents of parabolic induction

Let $C_i\in\operatorname{Comp}(\bm{d}_i)$ for $i=1,\dots,k$, let $V=V^1\oplus\cdots\oplus V^k$, and let $C\in\operatorname{Comp}(V)$. Let $\pi(C)$ denote the irreducible representation corresponding to $C$, let $\operatorname{JH}$ denote the set of Jordan–Hölder constituents, and let $p_i$ be the projection to the $i$th factor. Lapid–Mínguez's conjecture. The representation $\pi(C)$ occurs as a subquotient of $\pi(C_1)\times\cdots\times\pi(C_k)$ if and only if there is an irreducible component $D$ of $C\cap(C_1\times\cdots\times C_k)$ such that each restriction $p_i|_D$ is dominant onto $C_i$. In particular, if all but at most one $C_i$ are rigid, then

$$\operatorname{JH}(\pi(C_1)\times\cdots\times\pi(C_k))=\{\pi(C)\mid C\supseteq C_1\oplus\cdots\oplus C_k\},$$

and if all $C_i$ are rigid, then

$$\operatorname{JH}(\pi(C_1)\times\cdots\times\pi(C_k))=\{\pi(C)\mid x_1\oplus\cdots\oplus x_k\in C\},$$

where $x_i$ is a rigid element of $C_i$. The conjecture gives a geometric criterion for the representation-theoretic constituents of parabolic induction; the supplied text does not state a resolution.