Socle-filtration conjecture for highest derivatives

Let $\pi$ be an irreducible representation of $\mathrm{GL}_n$ of depth $d$. Regard its highest derivative $\pi^-$ as a representation of $M_{n-d+1}$, and let

$$0=F_0\pi^-\subset F_1\pi^-\subset\cdots\subset F_p\pi^- =\pi^-$$

be its increasing socle filtration, while

$$\pi^-=G^0\pi^-\supset G^1\pi^-\supset\cdots\supset G^q\pi^-=0$$

be its decreasing co-socle filtration. Socle-filtration conjecture. The following hold: (1) $p=q$ and $F_k\pi^-=G^{p-k}\pi^-$ for every integer $k$; (2) the filtration lengths are symmetric, namely

$$\operatorname{length}(F_k\pi^-/F_{k-1}\pi^-)=\operatorname{length}(F_{p-k+1}\pi^-/F_{p-k}\pi^-);$$

(3) $\pi^-$ has a unique irreducible quotient, and hence a unique irreducible submodule, equivalently $\operatorname{length}(F_1\pi^-)=1$. The conjecture concerns the structure of highest derivatives and is supported in the paper by computations; its general validity remains open.