Derivative formula for the representations $\rho_c(\tau)$

Let $\tau$ be a genuine essentially tempered generic representation of $\operatorname{GL}_k^{(m)}$, and let $\rho_c(\tau)$ be the associated representation of $\operatorname{GL}_{rkc}^{(m)}$. A representation is $(rk,c)$ when it has the corresponding specified highest-derivative structure.

Derivative conjecture. The representation $\rho_1(\tau)$ is $(rk,1)$. If $F$ is non-archimedean, then for every $c>1$ the highest derivative of $\rho_c(\tau)$ is

$$\rho_c^{(rk)}(\tau)=|\det|^{(r-1)/2}\rho_{c-1}(\tau).$$

This conjecture supplies the derivative input used to prove that the representations $\rho_c(\tau)$ are $(rk,c)$.