  We conjecture an explicit formula for the $K$-theoretically refined
Vafa-Witten invariants of the Enriques surface. By a wall-crossing argument the
conjecture is equivalent to a new conjectural formula for the K-theoretically
refined Pandharipande-Thomas invariants of the local Enriques surface. Evidence
for the conjecture is given in several cases. We also comment on the case of K3
surfaces previously studied by Thomas.
