The create/settings dialogs (salvaged from #114664) let a user set or clear
a board's project_id, but the binding was invisible outside Settings.
GET /boards already annotates every board with project_id + project_name,
so the switcher now renders a "Project: <name>" badge whose × sends
PATCH {project_id: ""} — the same clear the settings dialog uses — without
touching default_workdir.
Settings also stops sending default_workdir: "" next to a chosen project
when the directory field is blank: the explicit "" suppressed the server's
project → default_workdir mirror, so binding via Settings left the board
without the workspace default the create dialog would have seeded.
Trim the salvaged test to the invariants (selector wiring, payload shapes,
badge unbind) and document the control on the Kanban docs page.
0985aeb732