Stable linearisability conjecture for G-surfaces with permutation atoms

Let $G$ act on a surface $X$. An atom is a component in the atomic semi-orthogonal decomposition associated with the $G$-surface, and an atom is non-twisted permutation if it has the corresponding untwisted permutation form. Stable linearisability conjecture. If $X$ has only non-twisted permutation atoms, then $X$ is stably $G$-linearisable. The conjecture proposes that these categorical atoms detect stable equivariant linearisability. The source gives an example showing that ordinary $G$-linearisability need not follow, but does not resolve stable $G$-linearisability.