Finiteness conjecture for quasi-projective Fourier–Mukai partners

Let $X$ be a quasi-projective variety. A quasi-projective variety $X'$ is a Fourier–Mukai partner of $X$ if

$$\mathbf{D}^b(\operatorname{Coh} X')\cong\mathbf{D}^b(\operatorname{Coh} X).$$

Finiteness conjecture for Fourier–Mukai partners. If $X$ is a quasi-projective variety, then there are only finitely many quasi-projective Fourier–Mukai partners $X'$.

This conjecture expresses the expectation that the bounded derived category of coherent sheaves retains almost all information about a variety. The source provides no resolution status or further evidence for the general quasi-projective case.