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Anisotropic pair breaking close to surfaces favors the chiral $A$phase of the superfluid ${}^3\mathrm{He}$over the time-reversal invariant $B$phase. Confining the superfluid ${}^3\mathrm{He}$into a cavity of height $D$of the order of the Cooper pair size characterized by the coherence length ${\xi }_0$—ranging between 16 nm (34 bar) and 77 nm (0 bar)—extends the surface effects over the whole sample volume, thus allowing stabilization of the $A$phase at pressures $P$and temperatures $T$where otherwise the $B$phase would be stable. In this Letter, the surfaces of such a confined sample are covered with a superfluid ${}^4\mathrm{He}$film to create specular quasiparticle scattering boundary conditions, preventing the suppression of the superfluid order parameter. We show that the chiral $A$phase is the stable superfluid phase under strong confinement over the full $P\text{−}T$phase diagram down to a quasi-two-dimensional limit $D/{\xi }_0=1$, where $D=80\mathrm{nm}$. The planar phase, which is degenerate with the chiral $A$phase in the weak-coupling limit, is not observed. The gap inferred from measurements over the wide pressure range from 0.2 to 21.0 bar leads to an empirical ansatz for temperature-dependent strong-coupling effects. We discuss how these results pave the way for the realization of the fully gapped two-dimensional $p_x+i{p}_y$superfluid under more extreme confinement. Published by the American Physical Society2025