Equivalence between k-compact generation of stable oo-categories and k-well generation of their homotopy categories

Establish that a stable presentable oo-category E is k-compactly generated if and only if its homotopy category Ho(E) is k-well generated for the same regular cardinal k; specifically, prove the converse implication that Ho(E) being k-well generated implies that E is k-compactly generated (and, more generally, that Lemma A.8(b) holds for any presentable E).

Background

The authors prove one direction (Proposition A.10): if E is k-compactly generated, then Ho(E) is k-well generated. They point out the desirability of a full equivalence for the same k but could only establish one implication.

Such an equivalence would align cardinal-relative generation notions between stable oo-categories and their triangulated homotopy categories, streamlining the transfer of structural results across settings.