  We construct explicit elements $W_{ij}^k$ in (a completion of) the shifted
quantum toroidal algebra of type $A$, and show that these elements act by 0 on
the $K$-theory of moduli spaces of parabolic sheaves. We expect that the
quotient of the shifted quantum toroidal algebra by the ideal generated by the
elements $W_{ij}^k$ will be related to $q$-deformed $W$-algebras of type $A$
for arbitrary nilpotent, which would imply a $q$-deformed version of the AGT
correspondence between gauge theory with surface operators and conformal field
theory.
