Another counterexample to the Nerves of Steel Conjecture

In [B2] Balmer introduced the homological spectrum of a rigid tensor-triangulated category, and had the nerves of steel'' not to conjecture that it was in bijection with the prime spectrum, despite an avalanche of examples’'. Naturally, the statement came to be known as Balmer’s Nerves of Steel Conjecture. More recently, in [NVY] the conjecture was stated explicitly and extended to a more general non-symmetric context. In this paper we provide a counterexample to the original suggestion in [B2], so also to the conjecture from [NVY]. A first counterexample was recently constructed in [BHR].