  We study the (quantum) Schur algebras of type B/C corresponding to the Hecke
algebras with unequal parameters. We prove that the Schur algebras afford a
stabilization construction in the sense of Beilinson-Lusztig-MacPherson that
constructs a multiparameter upgrade of the quantum symmetric pair coideal
subalgebras of type A III/IV with no black nodes. We further obtain the
canonical basis of the Schur/coideal subalgebras, at the specialization
associated to any weight function. These bases are the counterparts of
Lusztig's bar-invariant basis for Hecke algebras with unequal parameters. In
the appendix we provide an algebraic version of a type D
Beilinson-Lusztig-MacPherson construction which is first introduced by Fan-Li
from a geometric viewpoint.
