  For any smooth complex projective surface $S$, we construct semistable
refined Vafa-Witten invariants of $S$ which prove the main conjecture of
arXiv:1810.00078. This is done by extending part of Joyce's universal
wall-crossing formalism to equivariant K-theory, and to moduli stacks with
symmetric obstruction theories, particularly moduli stacks of sheaves on
Calabi-Yau threefolds. An important technical tool which we introduce is the
symmetrized pullback, along smooth morphisms, of symmetric obstruction
theories.
