  For the recently discovered cuprate superconductor
$\mathrm{Ba_{2}CuO_{3+\delta}}$, we propose a lattice structure which resembles
the model considered by Lieb to represent the vastly oxygen-deficient material.
We first investigate the stability of the Lieb-lattice structure, and then
construct a multiorbital Hubbard model based on first-principles calculation.
By applying the fluctuation-exchange approximation to the model and solving the
linearized Eliashberg equation, we show that $s$-wave and $d$-wave pairings
closely compete with each other, and, more interestingly, that the
intra-orbital and inter-orbital pairings coexist. We further show that, if the
energy of the $d_{3z^2-r^2}$ band is raised to make it "incipient" with the
lower edge of the band close to the Fermi level within a realistic band filling
regime, $s\pm$-wave superconductivity is strongly enhanced. We reveal an
intriguing relation between the Lieb model and the two-orbital model for the
usual K$_2$NiF$_4$ structure where a close competition between $s-$ and
$d-$wave pairings is known to occur. The enhanced superconductivity in the
present model is further shown to be related to an enhancement found previously
in the bilayer Hubbard model with an incipient band.
