feat(dashboard): extract PR status loader and surface badges

Move pull-request discovery and caching into a dedicated background
loader component. The repository dashboard no longer performs the
network/keychain work itself and instead receives a shared badge state
that is refreshed and persisted independently. The app wires the loader
into top-level state and binds badge data into the dashboard view.

- Add a background loader to fetch, group and cache open PR counts.
- Simplify dashboard to accept and render pull-request badge state.
- Wire loader into app-level state and bind badges to the dashboard.

src/App.svelte

@@ -10,6 +10,9 @@
 
   import TitleBar from "./lib/TitleBar.svelte";
   import RepoToolbar from "./lib/RepoToolbar.svelte";
+  import PullRequestStatusLoader from "./lib/components/PullRequestStatusLoader.svelte";
+  import type { PullRequestBadgeState } from "./lib/pullRequestBadges";
+  let pullRequestBadges: Record<string, PullRequestBadgeState> = {};
   import RepositoryDashboard from "./lib/components/RepositoryDashboard.svelte";
   import ReviewCenter from "./lib/components/ReviewCenter.svelte";
   import RepoTabs from "./lib/RepoTabs.svelte";
@@ -5548,10 +5551,15 @@
       </section>
     {/if}
 
+    <PullRequestStatusLoader
+      repos={dashboardRepos} language={appLanguage}
+      integrations={gitIntegrationSettings} loadCredential={loadStoredCredential}
+      bind:pullRequestBadges
+    />
     {#if activeView === "management"}
       <RepositoryDashboard
         repos={dashboardRepos} language={appLanguage} {isBusy}
-        integrations={gitIntegrationSettings} loadCredential={loadStoredCredential}
+        {pullRequestBadges}
         onOpen={selectRepoTab} onAdd={chooseRepositoryFolder} onClone={openCloneDialog}
         onInit={initializeRepository}
         onFavorite={toggleFavoriteRepo} onClose={closeDashboardRepository}