The Fourier–Mukai support-dimension invariance conjecture

For a smooth projective variety $X$, let $N_X$ denote its Fourier–Mukai support dimension. Two smooth projective varieties are Fourier–Mukai partners if they are related by a Fourier–Mukai equivalence.

Fourier–Mukai support-dimension invariance conjecture. If smooth projective varieties $X$ and $Y$ are Fourier–Mukai partners, then

$$N_X=N_Y.$$

The same conjectural statement also asserts that if $X$ is of $K$-equivalent type and there is a Fourier–Mukai transform from $X$ to $Y$, then that transform is of $K$-equivalent type. The parser reports this candidate as disproved; Kawamata's related prediction that birationally equivalent, derived-equivalent smooth projective varieties are $K$-equivalent was counterexampled by the author.