On a conjecture of Ayala–Francis–Rozenblyum

I will present some finite stratified homotopy types constructed from smooth data which are not the homotopy types of conically smooth manifolds, disproving a 2015 conjecture of Ayala, Francis and Rozenblyum. The main tool is a new combinatorial and fully faithful exit path infinity-category construction. Time permitting, I will discuss a future direction for such conjectures.