  In this paper we study the contribution of monopole bubbling to the
expectation value of supersymmetric 't Hooft defects in Lagrangian theories of
class $\mathcal{S}$ on $\mathbb{R}^3\times S^1$. This can be understood as the
Witten index of an SQM living on the world volume of the 't Hooft defect that
couples to the bulk 4D theory. The computation of this Witten index has many
subtleties originating from a continuous spectrum of scattering states along
the non-compact vacuum branches. We find that even after properly dealing with
the spectral asymmetry, the standard localization result for the 't Hooft
defect does not agree with the result obtained from the AGT correspondence. In
this paper we will explicitly show that one must correct the localization
result by adding an extra term to the standard Jeffrey-Kirwan residue formula.
This extra term accounts for the contribution of ground states localized along
the non-compact branches. This extra term restores both the expected symmetry
properties of the line defect expectation value and reproduces the results
derived using the AGT correspondence.
