  We solve the Dirichlet problem for Monge-Amp\`ere equation for $(n-1)$-PSH
functions possibly with degenerate right-hand side, through deriving a
quantitative version of boundary estimate under the assumption of $(n-1)$-PSH
subsolutions. In addition, we confirm the subsolution assumption on a product
of a closed balanced manifold with a compact Riemann surface with boundary.
