Do color-neutral top partners, as in Twin Higgs models, exist?

In neutral-naturalness models (Twin Higgs, folded supersymmetry, quirky little Higgs) the top-loop contribution to $m_h^{2}$ is cancelled by states without SM color, with a mirror-sector scale f; Higgs couplings shift by about $v^{2}/(2 f^{2})$ and the Higgs can decay to mirror glueballs with displaced vertices (approximate relations, model dependent). Determine whether such a sector exists with $f\ \text{below about}\ 1\ \text{to}\ 2\ \mathrm{TeV}$, by measuring $\mathrm{BR}(h \to \text{mirror hadrons})$ down to about $10^{-4}$ and Higgs couplings at the per-mille level, or exclude it.

Displaced-vertex searches for exotic Higgs decays at the HL-LHC and at dedicated long-lived-particle detectors, together with per-mille Higgs coupling measurements at a Higgs factory.