Wild blocks of type $A$ Hecke algebras are strictly wild

We prove that all wild blocks of type $A$ Hecke algebras with quantum characteristic $e⩾3$ – i.e. blocks of weight at least $2$ – are strictly wild, with the possible exception of the weight $2$ Rouquier block for $e=3$. As a corollary, we show that for $e⩾3$, all wild blocks of the $q$-Schur algebras are strictly wild, without exception.