Multiplicativity conjecture for the Bernstein–Zelevinsky derivative

Let $G_n$ denote the relevant general linear group, and let $\mathcal{M}_{\infty}^{d_i}(G_{n_i})$ be the class of smooth representations of degree $d_i$ for $i=1,2$. Set

$$n=n_1+n_2,$$

and let $\pi_i\in\mathcal{M}_{\infty}^{d_i}(G_{n_i})$. Define $\pi:=\pi_1\times\pi_2$ and $d:=d_1+d_2$, so that $\pi\in\mathcal{M}_{\infty}^{d}(G_n)$. Multiplicativity conjecture. The Bernstein–Zelevinsky derivative satisfies

$$E_{\infty}^{d}(\pi)=E_{\infty}^{d_1}(\pi_1)\times E_{\infty}^{d_2}(\pi_2).$$

This conjectures a generalization of the product formula previously established for monomial representations. The parser marks the question as open; the paper presents it among its open questions.