Balmer's Nerves of Steel Conjecture

Let $\cat{C}$ be a rigid $tt$-category. Its comparison map

$$\phi\colon \Spc^h(\cat C) \twoheadrightarrow \Spc(\cat C)$$

Balmer's Nerves of Steel Conjecture. The map $\phi$ is bijective. This conjecture seeks to reconcile the homological and triangular spectra of a rigid tensor-triangulated category. The source explains that the comparison map is known to be surjective and that the conjecture holds in the classical stable homotopy example, but does not provide a general resolution.