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We show that steady-state expectation values predicted by the universal Lindblad equation (ULE) are accurate up to bounded corrections that scale linearly with the effective system-bath coupling, $Γ$ (second order in the microscopic coupling). We also identify a near-identity, quasilocal "memory-dressing" transformation, used during the derivation of the ULE, whose inverse can be applied to achieve relative deviations of observables that generically scale to zero with $Γ$, even for nonequilibrium currents whose steady-state values themselves scale to zero with $Γ$. This result provides a solution to recently identified limitations on the accuracy of Lindblad equations, which highlighted a potential for significant relative errors in currents of conserved quantities. The transformation we identify allows for high-fidelity computation of currents in the weak-coupling regime, ensuring thermodynamic consistency and local conservation laws, while retaining the stability and physicality of a Lindblad-form master equation.