Proper Fourier–Mukai locus conjecture for triangulated categories

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

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

In particular,

$$\dim\operatorname{Spec}^{\mathsf{Ser}}\mathcal T=\dim\operatorname{Spec}^{\mathsf{pFM}}\mathcal T=\dim X,$$

including the last equality as part of the conjecture. The source presents this as an a priori weaker conjecture because proper varieties need not be projective in higher dimension; it gives no resolution.