Equality of genuine and naive homological support

Prove or disprove that the genuine homological support $cSupph(t)$ equals the naive homological support $cSupphnaive(t)$ for every object $t$ in every rigidly-compactly generated tensor-triangulated category.

Background

The paper establishes the inclusion of genuine homological support in naive homological support and records equality for several important classes of objects, including compact objects, weak rings, and weak corings. It then identifies the general equality as an explicit open question. A positive answer would show that the two support theories agree universally.

References

It is an open question whether~eq:supph-inclusion is always an equality for all objects~$t\in\cat T$.

eq:supph-inclusion:

$\Supph(t) \subseteq \Supphnaive(t) $

— Local Bousfield classes via homological support  (2608.26876 - Barthel et al., 27 Aug 2026) in Remark 2.10, Section 2, “Classifying homological Bousfield classes”