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The rekey suite's Phase-1 gate fails the whole run when one pair is still unreachable at the 65s cap, even though the strict all-pairs assertion that follows is the real test. On slower runners a mesh that sat at 19/20 for most of the window was reported as a convergence failure rather than reaching the assertion that would say what was wrong. wait_until_connected gains an optional sixth argument, near_converged_accept_secs, default 0. It acts only at the hard cap: if the current near-converged hold (within slack, no progress for stall_secs) has lasted at least that long, the gate returns 0 with the new verdict near_converged instead of timing out. Progress disarms the hold, so only the hold that ends at the cap counts. Acceptance is reachable only in states that were red before, so it cannot turn a run that would have converged by the cap into a failure, and the strict assertion still decides. Only the rekey Phase-1 gate passes the argument (10s). Every other caller passes five or fewer arguments and is unchanged. The hermetic gate test gains case 7 covering acceptance, the default of 0, stalls beyond slack, holds shorter than the window, late convergence before the cap, and the hold's re-arm and reset on progress.