  We provide a new extension to the geometric construction of 6d $(1,0)$ SCFTs
that encapsulates Higgs branch structures with identical global symmetry but
different spectra. In particular, we find that there exist distinct 6d $(1,0)$
SCFTs that may appear to share their tensor branch description, flavor symmetry
algebras, and central charges. For example, such subtleties arise for the very
even nilpotent Higgsing of $(\mathfrak{so}_{4k}, \mathfrak{so}_{4k})$ conformal
matter; we propose a method to predict at which conformal dimension the Higgs
branch operators of the two theories differ via augmenting the tensor branch
description with the Higgs branch chiral ring generators of the building block
theories. Torus compactifications of these 6d $(1,0)$ SCFTs give rise to 4d
$\mathcal{N}=2$ SCFTs of class $\mathcal{S}$ and the Higgs branch of such 4d
theories are captured via the Hall--Littlewood index. We confirm that the
resulting 4d theories indeed differ in their spectra in the predicted conformal
dimension from their Hall--Littlewood indices. We highlight how this ambiguity
in the tensor branch description arises beyond the very even nilpotent Higgsing
of $(\mathfrak{so}_{4k}, \mathfrak{so}_{4k})$ conformal matter, and hence
should be understood for more general classes of 6d $(1,0)$ SCFTs.
