The salvaged read-side fix lets a profile resolve the xAI OAuth grant from
the global-root auth store when it has no own providers.xai-oauth block.
But _save_xai_oauth_tokens still wrote rotated tokens only to the active
profile store. Because xAI rotates the refresh_token on every refresh, a
profile that reads root's grant and refreshes it left root holding a now-
revoked refresh token — killing every other profile reading the stale root
grant with invalid_grant once its access token expired (#43589).
Detect the read-from-root case (profile lacks its own providers.xai-oauth
block) and, after the profile save, write the rotated chain back to the
global root too via a best-effort, TOCTOU-safe write-through that reuses
_save_auth_store with an explicit target path. A profile that genuinely
shadows root (has its own block) is left untouched, classic mode is a
no-op, and a failed root write never breaks the profile's own save.
Pairs with the read fallback in the preceding commit so the cross-profile
xAI grant stays coherent in both directions.
