Balmer's homological-to-tensor triangular spectrum bijection conjecture

Let $\cat K$ be an essentially small tensor triangulated category. Write $\Spch(\cat K)$ for its homological spectrum, $\Spc(\cat K)$ for its Balmer spectrum, and

$$\phi\colon \Spch(\cat K) \to \Spc(\cat K)$$

for the canonical continuous comparison map. Balmer's conjecture. The map $\phi$ is always a bijection.

The conjecture asks whether homological support and tensor triangular support have the same underlying spectrum. The comparison map is surjective under very mild hypotheses and is bijective in all known examples, but the general assertion remains open.