We study centralized two-way dynamic matching markets in which agents arrive stochastically over time and wait in queues associated with their types. We focus on static priority policies that assign each agent type a fixed ranking over its neighbors and make online matching decisions using only queue availability. We propose the General Priority ($\mathbf{GP}$) policy, which builds on the Tree Priority ($\mathbf{TP}$) policy introduced in [KAG25]. Under the general position gap condition, $\mathbf{GP}$ selects matches using a nondegenerate optimal basic solution of the fluid relaxation. It prioritizes children over parents within trees and consistently gives clockwise neighbors priority over counterclockwise neighbors on each cycle. We establish the optimal all-time regret scaling $O(ε^{-1})$ for $\mathbf{GP}$ on general two-way matching networks, where the general position gap $ε$ measures the fluid relaxation's distance from degeneracy. In particular, this resolves the conjecture by [KAG25] on the optimal $ε$-dependence for $\mathbf{TP}$.

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