[obaseyi2]

Fixes #618.

The tab bar keeps the previous DOM index as its keyboard entry point after selection/reordering. Arrow-key browsing can also leave an unselected tab as the entry point after focus exits.

Synchronize tabindex after rendering and focus transitions: use the selected tab when focus is outside, preserve the focused item while navigating inside, and retain first-tab/add-button fallbacks. This changes neither selection nor focus automatically. Changes are in tabbar.ts and its tests.

Reproduction

1. Create three tabs, select the third, and focus a button before the bar.
2. Press Tab: focus should enter the third tab.
3. Press Left Arrow to browse the second tab without selecting it.
4. Press Tab to leave, then Shift+Tab: focus should return to the selected third tab.
5. Move the selected title to the first position and repeat entry: focus should follow that title.

Validation

• Widget suite: 984 tests passed in each of Chromium and Firefox.
• Original source failed six regression tests in Chromium.
• Eight native-key checks passed in each browser, including DockPanel content traversal and reorder.
• Widgets build/bundling, changed-file lint/format and diff whitespace checks passed.

WebKit and full JupyterLab were not tested.