Socle formulas for rigid irreducible components of Lusztig's nilpotent varieties in type $A$

Let $F$ be a non-archimedean local field. For any irreducible component $C$ of a nilpotent variety of type $A$ let $\pi (C)$ be the irreducible representation of a general linear group over $F$ attached to it. We show that if the multisegment of $C$ is multiplicity free, $C$ is rigid if and only if $\pi (C)$ is $\mathrm{\square }$-irreducible. In this case, $$\pi (C\ast D)=soc(\pi (C)\times \pi (D)),\pi (D\ast C)=soc(\pi (D)\times \pi (C))$$ for every irreducible component $D$, where $\ast$ is the binary operation defined by Aizenbud and Lapid, in accordance with a conjecture of Lapid and Mínguez. We also give a computer-assisted example in $A_7$ of a non-simple rigid module $z\mathrm{\ne }0$ which does not admit a proper rigid submodule $y\mathrm{\ne }0$ such that $Ex{t}_{\mathrm{\Pi }}^1(z,z\mathrm{/}y)=0$, answering negatively a question of Aizenbud and Lapid.