  The infrared formula relates the Schur index of a 4d $\mathcal{N}=2$ theory
to its wall-crossing invariant, a.k.a. BPS monodromy. A further extension of
this formula, proposed by C\'ordova, Gaiotto and Shao, includes contributions
by various types of line and surface defects. We study BPS monodromies in the
presence of vortex surface defects of arbitrary vorticity for general class
$\mathcal{S}$ theories of type $A_1$ engineered by UV curves with at least one
regular puncture. The trace of these defect BPS monodromies is shown to
coincide with the action of certain $q$-difference operators acting on the
trace of the (pure) 4d BPS monodromy. We use these operators to develop a
"bootstrap" (of traces) of BPS monodromies, relying only on their infrared
properties, thereby reproducing the very general ultraviolet characterization
of the Schur index.
