Topological structure conjecture for the Serre-invariant locus

Let $\mathcal T$ be a triangulated category with $X\in\operatorname{FM}\mathcal T$. A topological space is locally spectral if it has an open cover by spectral spaces, and locally noetherian if it is locally noetherian. Serre-invariant locus structure conjecture. The space $\operatorname{Spec}^{\mathsf{Ser}}\mathcal T$ is locally spectral, hence sober, and locally noetherian of dimension $\dim X$. Moreover, its connected components and irreducible components coincide and are isomorphic to one another. The source derives this as a consequence expected if the main Serre-invariant locus conjecture holds; no independent resolution is given.