Birationality conjecture for geometrically rational surfaces with the same atoms

Let $X$ and $Y$ be geometrically rational surfaces, and let nontrivial atoms mean the atoms other than the trivial pieces in their atomic semi-orthogonal decompositions. Birationality conjecture for geometrically rational surfaces. The surfaces $X$ and $Y$ are birational if and only if they have the same nontrivial atoms. This extends the proved birationality criterion for birationally rich surfaces and would give a categorical criterion for birationality among all geometrically rational surfaces. The source identifies remaining cases among minimal del Pezzo surfaces and Mori conic bundles of degree at most $4$, so the conjecture remains open.