  We conjecture a formula for the refined $\mathrm{SU}(3)$ Vafa-Witten
invariants of any smooth surface $S$ satisfying $H_1(S,\mathbb{Z}) = 0$ and
$p_g(S)>0$. The unrefined formula corrects a proposal by Labastida-Lozano and
involves unexpected algebraic expressions in modular functions. We prove that
our formula satisfies a refined $S$-duality modularity transformation.
  We provide evidence for our formula by calculating virtual $\chi_y$-genera of
moduli spaces of rank 3 stable sheaves on $S$ in examples using Mochizuki's
formula. Further evidence is based on the recent definition of refined
$\mathrm{SU}(r)$ Vafa-Witten invariants by Maulik-Thomas and subsequent
calculations on nested Hilbert schemes by Thomas (rank 2) and Laarakker (rank
3).
