Let $X_0$ be a smooth, geometrically connected scheme over $\mathbb{F}_q$. As usual, let $\tau : \bar{\mathbb{Q}}_{\ell} \simeq \mathbb{C}$ be a fixed isomorphism. Let $\mathcal{C}$ be the category of smooth Weil sheaves on $X_0$, with its rank function $\mathrm{rk} : \operatorname{ob} \mathcal{C} \rightarrow \mathbb{N}$. An object $\mathcal{L}$ of $\mathcal{C}$ of rank $1$ is always $\tau$-pure, and has a well defined $\tau$-weight $\omega(\mathcal{L})$.
If $\mathcal{F}$ is any object of $\mathcal{C}$, its degree $\deg\mathcal{F} = \omega ( \det \mathcal{F} )$ is defined as the $\tau$-weight of its maximal exterior power. These functions $\deg$ and $\mathrm{rk}$ define a Harder-Narasimhan filtration on any object of $\mathcal{C}$ (cf this article for example).
Is the tensor product of two semistable smooth Weil sheaves always semistable?
It follows from the discussion of determinant weights in Weil II that this is indeed the case if the two sheaves are supposed to be geometrically semisimple (and that under this assumption, "semistable = isoclinic"). What about the general case?
