fix(utils): display_path normalizes home-relative suffix to platform separator

`display_path_with_home` joined the `~` prefix with `MAIN_SEPARATOR_STR`
but called `rest.display()` for the suffix, which preserves whatever
separators the input carried. On Windows that produced mixed-separator
output like `~\projects/foo` for any path that came in with forward
slashes — visible in the tests that #506 added to lock down the
contract (the tests passed locally on Unix but failed on the
windows-latest CI runner).

Walk `rest.components()` and join each `Normal` component with
`MAIN_SEPARATOR_STR`. Pure-Rust, no extra deps, behavior is
byte-identical on Unix because the input separator was already `/`.

Verified locally:
- `cargo test -p deepseek-tui --locked display_path` ✓ (5 passed)

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

crates/tui/src/utils.rs

@@ -344,6 +344,10 @@ pub fn display_path(path: &Path) -> String {
 /// Like [`display_path`] but takes an explicit home directory instead of
 /// reading `$HOME` / `dirs::home_dir()`.  Used in tests and anywhere the
 /// caller already has the home path available.
+///
+/// The home-relative suffix is rejoined with the platform separator
+/// (`\` on Windows, `/` elsewhere) by walking the path's components, so
+/// inputs that carried foreign separators don't leak through.
 #[must_use]
 pub fn display_path_with_home(path: &Path, home: Option<&Path>) -> String {
     let Some(home) = home else {
@@ -353,8 +357,13 @@ pub fn display_path_with_home(path: &Path, home: Option<&Path>) -> String {
         if rest.as_os_str().is_empty() {
             return "~".to_string();
         }
-        let sep = std::path::MAIN_SEPARATOR;
-        return format!("~{sep}{}", rest.display());
+        let sep = std::path::MAIN_SEPARATOR_STR;
+        let mut out = String::from("~");
+        for component in rest.components() {
+            out.push_str(sep);
+            out.push_str(&component.as_os_str().to_string_lossy());
+        }
+        return out;
     }
     path.display().to_string()
 }