Hille–Perling's standard augmentation conjecture for strong exceptional collections

Let $X$ be a rational surface. A standard augmentation is a toric system obtained from a toric system on a Hirzebruch surface by successive blow-ups and the corresponding augmentation operations. Two line bundles are completely orthogonal when the relevant Ext groups vanish in both directions; a permutation of a collection means reordering within such mutually orthogonal blocks. Hille–Perling's conjecture. Every full strong exceptional collection of line bundles on $X$ corresponds, up to permutation of completely orthogonal bundles, to a standard augmentation. A positive answer would describe all full exceptional collections of line bundles on rational surfaces, but the supplied text does not establish the conjecture.