Serre-invariant locus conjecture for triangulated categories

Let $\mathcal T$ be a triangulated category with $X \in \operatorname{FM} \mathcal T$. The Fourier–Mukai locus $\operatorname{Spec}^{\mathsf{FM}} \mathcal T$ and the Serre-invariant locus $\operatorname{Spec}^{\mathsf{Ser}} \mathcal T$ are the corresponding loci in the triangular spectrum. Serre-invariant locus conjecture.

$$\operatorname{Spec}^{\mathsf{FM}} \mathcal T=\operatorname{Spec}^{\mathsf{Ser}} \mathcal T.$$

In particular,

$$\dim \operatorname{Spec}^{\mathsf{Ser}} \mathcal T=\dim \operatorname{Spec}^{\mathsf{FM}} \mathcal T=\dim X.$$

The conjecture extends equality known for curves and varieties with ample or anti-ample canonical bundle; the source notes an additional quiver example but gives no general resolution.