Variation of Weyl modules in $p$-adic families

Given a Weil-Deligne representation with coefficients in a domain, we prove the rigidity of the structures of the Frobenius-semisimplifications of the Weyl modules associated to its pure specializations. Moreover, we show that the structures of the Frobenius-semisimplifications of the Weyl modules attached to a collection of pure representations are rigid if these pure representations lift to Weil-Deligne representations over domains containing a domain $\mathscr{O}$ and a pseudorepresentation over $\mathscr{O}$ parametrizes the traces of these lifts.