  In the search for transverse-universal knots in the standard contact
structure of $\mathbb{S}^3$, we present a classification of the transverse
twist knots with maximal self-linking number, that admit only overtwisted
contact branched covers. As a direct consequence, we obtain an infinite family
of transverse knots in $(\mathbb{S}^3,\xi_{std})$ that are not
transverse-universal, although they are universal in the topological sense.
