fix(desktop): match a project to its cwd across Windows path spellings

Project ownership compared paths literally, so a nested cwd failed to match
its project whenever the separator or drive-letter case differed — which on
Windows is routine. Normalise both sides for comparison only, folding case
for drive and UNC paths.

apps/desktop/src/store/projects.test.ts

@@ -22,6 +22,7 @@ import {
   exitProjectScope,
   openProjectCreate,
   pickProjectFolder,
+  projectIdForCwd,
   projectNameForCwd,
   refreshProjects,
   refreshProjectTree,
@@ -241,6 +242,13 @@ describe('projectNameForCwd', () => {
     expect(projectNameForCwd('/elsewhere/mono-feature/src')).toBe('Monorepo')
   })
 
+  it('matches nested Windows paths across separator and case differences', () => {
+    $projectTree.set([treeNode({ id: 'p_win', label: 'Windows app', path: 'C:\\Repos\\App' })])
+
+    expect(projectIdForCwd('c:/repos/app/src')).toBe('p_win')
+    expect(projectNameForCwd('c:/repos/app/src')).toBe('Windows app')
+  })
+
   it('ignores auto-projects and the No-project bucket (no named identity)', () => {
     $projectTree.set([
       treeNode({ id: '/repos/loose', label: 'loose', path: '/repos/loose', isAuto: true }),

apps/desktop/src/store/projects.ts

@@ -229,8 +229,26 @@ export function resolveNewSessionCwd(): string {
   return workspaceCwdForNewSession()
 }
 
-const underPath = (parent: string, child: string): boolean =>
-  child === parent || child.startsWith(parent.endsWith('/') ? parent : `${parent}/`)
+// Comparison-only path spelling: accept either separator everywhere, trim a
+// trailing separator, and fold case for Windows drive/UNC paths. The backend
+// uses the same host-aware identity rule when assigning sessions to projects.
+const comparisonPath = (path: string): string => {
+  const raw = path.trim()
+  const windows = /^[A-Za-z]:[/\\]/.test(raw) || raw.startsWith('\\') || raw.startsWith('//')
+  const normalized = raw.replace(/\\/g, '/').replace(/\/+$/, '')
+
+  return windows ? normalized.toLowerCase() : normalized
+}
+
+const underPath = (parent: string, child: string): boolean => {
+  const normalizedParent = comparisonPath(parent)
+  const normalizedChild = comparisonPath(child)
+
+  return (
+    normalizedChild === normalizedParent ||
+    normalizedChild.startsWith(normalizedParent.endsWith('/') ? normalizedParent : `${normalizedParent}/`)
+  )
+}
 
 // The project (explicit or auto) that owns `cwd`, by longest path match across
 // the live tree. Null when no project covers it (it'll surface as a fresh