Outside of a drag the store is the source of truth (cross-tab
updates flow in for free). During a drag we keep splits in local
state so mousemove does not round-trip through the store. On
mouseup we persist the final splits via setPref. Removes
document.cookie reads/writes — cookieKey is reused as the
store-key for backwards compat.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>