  Nonperturbative inequalities constrain the pressure of Quantum Chromodynamics
(QCD) with its phase-quenched version, a Sign-Problem-free theory amenable to
lattice treatment. In the perturbative regime with a small QCD coupling
constant $\alpha_s$, one of these inequalities manifests as an $O(\alpha_s^3)$
difference between the phase-quenched and QCD pressures at large baryon
chemical potential $\mu_\text{B}$. In this work, we generalize state-of-the-art
algorithmic techniques in vacuum quantum field theory to address large-scale
multiloop computations at finite chemical potential. Using this approach, we
evaluate this $O(\alpha_s^3)$ difference and show that it is a
gauge-independent and small positive number compared to the known perturbative
coefficients at this order. This implies that phase-quenched lattice
simulations can provide a complementary nonperturbative method for accurately
determining the pressure of cold and dense quark matter at $O(\alpha_s^3)$.
