To determine if the endomorphism ring of a projective-injective module is symmetric is of wide interest in representation theory, ring theory and algebraic Lie theory. In this talk we show that for any projective-injective module over a large class of algebras which includes all quasi-hereditary truncations and quotients of (quantised, cyclotomic) Schur algebras and certain quotients of Hecke algebras of type $A$ that the endomorphism algebra is symmetric. We present an explicit example showing that endomorphism algebras of projective-injective modules in the BGG category $\mathcal{O}_q$ for the quantum group $U_q$ need not be symmetric, thereby disproving a conjecture by Andersen and Mazorchuk. This talk is based on a joint work with Fang Ming.