  The Higgs branch of 4d $\mathcal{N}=2$ SCFTs can be analyzed via the Hilbert
series of the Higgs branch or, in special cases, by computing the
Hall-Littlewood index. For any class $\mathcal{S}$ theory corresponding to a
genus-zero Riemann surface, they are conjectured to be identical. We present
several families of counterexamples. We find that for any class $\mathcal{S}$
theory with four or more $\mathbb{Z}_2$-twisted punctures, they do not match.
We construct 3d mirrors for such theories and analyze their Coulomb branch
Hilbert series to compute the Higgs branch Hilbert series of the 4d theory. We
further construct $a=c$ theories in class $\mathcal{S}$ using the twisted
punctures, and these theories, which includes the $\hat{D}_4(SU(2n+1))$
theories, have Hall--Littlewood index different from the Hilbert series of the
Higgs branch. We conjecture that this is the case for all $a=c$ theories with
non-empty Higgs branch, including $\mathcal{N}\ge 3$ SCFTs.
