  We study the integrable two-species asymmetric simple exclusion process
(ASEP) for two inequivalent types of open, non particle conserving boundary
conditions. Employing the nested off-diagonal Bethe ansatz method, we construct
for each case the corresponding homogeneous $T$-$Q$ relations and obtain the
Bethe ansatz equations. Numerical checks for small system sizes show
completeness for some Bethe ansatz equations, and partial completeness for
others.
