Add a runtime self-test page that exercises both paths

[mogul]

Restored from slopsmith/slopsmith-plugin-splitscreen#46 — original issue, opened by @byrongamatos on 2026-05-09.
[restored-from: slopsmith/slopsmith-plugin-splitscreen#46]

From specs/001-splitscreen/tasks.md — generated retrospectively by spec-kit.

Add a runtime self-test page that exercises both paths.

---

See specs/001-splitscreen/spec.md and specs/001-splitscreen/analyze.md for context.
This issue was filed in bulk; close as wontfix if no longer relevant.