Equality between Serre dimension and the infimum of global dimension

Determine whether the inequality \(\operatorname{Sdim}(\mathcal D)\leq \operatorname{Gd}(\mathcal D)\) is an equality for every triangulated category \(\mathcal D\) admitting a nonempty stability space, and establish a general equality theorem or produce a counterexample.

Background

For a triangulated category with nonempty stability space, Kikuta–Ouchi–Takahashi established the lower bound $$\operatorname{Sdim}(\mathcal D)\leq \operatorname{Gd}(\mathcal D)$$, where $\operatorname{Sdim}$ denotes the upper Serre dimension and $\operatorname{Gd}$ is the infimum of the global dimension function over stability conditions.

The paper proves equality for all smooth projective varieties and for the homologically smooth graded gentle categories considered in Example 5.1. Nevertheless, the authors state that no theorem establishes equality in general and that no counterexample is currently known, leaving the general relationship unresolved.