The project pill is first in the composer footer, preselected from the
URL (/p/shop-frontend/new). Picking a different project swaps the whole composer
scope — the skill/workflow picker, /-autocomplete, runner and base-branch pills all
reload from /api/p/<project>/…, the draft is kept per project
(cez-new-task-draft:<projectId>), and Start posts to
POST /api/p/<project>/runs.