  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.
