perf(plex): a card fetches its picture at a card's size, not the detail page's

Cards fetched the full cuts: a 400x600 poster (~38 KB) for a ~144px picture and the 1280x720
backdrop (~162 KB) for a ~212px shelf picture. The image proxy now also cuts thumb_sm (240x360)
and art_sm (480x270), cached apart; card pictures (grid, shelves, collection tiles, Up next) carry
both in srcset with sizes=auto, so the browser picks by laid-out width and DPR. Localdev:
17 KB a poster, 56 KB a backdrop. The detail page keeps the full cuts.

backend/tests/test_plex_image_sizes.py

@@ -0,0 +1,77 @@
+"""A card asks the Plex image proxy for a small cut; a detail page keeps the full one (E31 S4e).
+
+Cards fetched the 400×600 poster for a ~144px picture and the 1280×720 backdrop for a ~212px shelf
+picture. The `_sm` variants are cut by Plex at a card's size and cached apart from the full ones; an
+unknown variant still gets the full poster. DB lane (the proxy opens its own short session).
+"""
+
+from fastapi.testclient import TestClient
+
+import app.state as app_state
+from app.main import app
+from app.models import PlexItem, PlexLibrary
+from app.plex import imgcache
+from app.routes.plex import playback
+
+
+class _FakePlex:
+    calls: list[tuple[str, int | None, int | None]] = []
+
+    def __init__(self, db) -> None:
+        pass
+
+    def __enter__(self):
+        return self
+
+    def __exit__(self, *exc) -> None:
+        return None
+
+    def image_bytes(self, key, width=None, height=None):
+        _FakePlex.calls.append((key, width, height))
+        return b"\xff\xd8jpeg", "image/jpeg"
+
+
+def _setup(db, make_user, monkeypatch, tmp_path) -> TestClient:
+    user = make_user()
+    lib = PlexLibrary(plex_key="90", title="Movies", kind="movie", enabled=True)
+    db.add(lib)
+    db.flush()
+    db.add(
+        PlexItem(
+            rating_key="pm1",
+            library_id=lib.id,
+            kind="movie",
+            title="A Movie",
+            thumb_key="/library/metadata/pm1/thumb/1",
+            art_key="/library/metadata/pm1/art/1",
+        )
+    )
+    db.commit()
+    monkeypatch.setattr(app_state, "_configured_cache", True)
+    monkeypatch.setattr(playback, "resolved_user_id", lambda request: (user.id, True))
+    monkeypatch.setattr(playback, "PlexClient", _FakePlex)
+    monkeypatch.setattr(imgcache, "CACHE_DIR", tmp_path)
+    _FakePlex.calls = []
+    return TestClient(app)
+
+
+def test_each_variant_is_cut_at_its_own_size(db, make_user, monkeypatch, tmp_path):
+    client = _setup(db, make_user, monkeypatch, tmp_path)
+    for variant in ("thumb", "thumb_sm", "art", "art_sm"):
+        assert client.get(f"/api/plex/image/pm1?variant={variant}").status_code == 200
+    assert _FakePlex.calls == [
+        ("/library/metadata/pm1/thumb/1", 400, 600),
+        ("/library/metadata/pm1/thumb/1", 240, 360),
+        ("/library/metadata/pm1/art/1", 1280, 720),
+        ("/library/metadata/pm1/art/1", 480, 270),
+    ]
+    # Each cut is cached under its own name: asking again reaches Plex no more.
+    for variant in ("thumb", "thumb_sm", "art", "art_sm"):
+        client.get(f"/api/plex/image/pm1?variant={variant}")
+    assert len(_FakePlex.calls) == 4
+
+
+def test_an_unknown_variant_gets_the_full_poster(db, make_user, monkeypatch, tmp_path):
+    client = _setup(db, make_user, monkeypatch, tmp_path)
+    assert client.get("/api/plex/image/pm1?variant=huge").status_code == 200
+    assert _FakePlex.calls == [("/library/metadata/pm1/thumb/1", 400, 600)]