Families over the integral Bernstein Center and Tate cohomology of local Base change lifts for GL(n, F)

Let $p$ and $l$ be distinct odd primes, and let $F$ be a $p$-adic field. Let $\pi$ be a generic smooth integral representation of ${\rm GL}_n(F)$ over an $\overline{\mathbb{Q}}_l$-vector space. Let $E$ be a finite Galois extension of $F$ with $[E:F]=l$. Let $\Pi$ be the base change lift of $\pi$ to the group ${\rm GL}_n(E)$. Let $\mathbb{W}^0(\Pi, \psi_E)$ be the lattice of $\overline{\mathbb{Z}}_l$-valued functions in the Whittaker model of $\Pi$, with respect to a standard ${\rm Gal}(E/F)$-equivaraint additive character $\psi_E:E\rightarrow \overline{\mathbb{Q}}_l^\times$. We show that the unique generic sub-quotient of the zero-th Tate cohomology group of $\mathbb{W}^0(\Pi, \psi_E)$ is isomorphic to the Frobenius twist of the unique generic sub-quotient of the mod-$l$ reduction of $\pi$. We first prove a version of this result for a family of smooth generic representations of ${\rm GL}_n(E)$ over the integral Bernstein center of ${\rm GL}_n(F)$. Our methods use the theory of Rankin-selberg convolutions and simple identities of local $\gamma$-factors. The results of this article remove the hypothesis that $l$ does not divide the pro-order of ${\rm GL}_{n-1}(F)$ in our previous work.
Comment: Preliminary version, comments are welcome