Ext-enhanced monoidal Koszul duality for ${\mathrm{G}\mathrm{L}}_2$

The Hecke category participates in an equivalence called monoidal Koszul duality, which exchanges it with the category of (Langlands-dual) “free-monodromic tilting sheaves.” Motivated by a recent conjecture of Gorsky and the first-named author on HOMFLYPT link homology, we propose to enhance this duality with an additional grading. We provide evidence for this enhancement in the case of ${\mathrm{G}\mathrm{L}}_2$, working in the language of the second-named author’s joint work with Achar, Riche, and Williamson.