The binary-operation conjecture for representations of $GL_n$

Let $C_1$ and $C_2$ be irreducible components of Lusztig's nilpotent varieties, let $*$ denote the binary operation on such components, and let $\pi(C)$ be the representation associated with an irreducible component $C$. A representation is $\square$-irreducible when its relevant product with itself is irreducible.

Binary-operation conjecture. $\pi(C_1*C_2)$ is a subrepresentation of $\pi(C_1)\times\pi(C_2)$. In particular, if $\pi(C_1)$ or $\pi(C_2)$ is $\square$-irreducible, then

$$\pi(C_1*C_2)=\operatorname{soc}(\pi(C_1)\times\pi(C_2)).$$

This conjecture is presented as an application of the binary operation on irreducible components and is inspired by an earlier conjecture. Its resolution is not specified in the supplied text.