[Sync] Refactor non-frontend DTC to handle new API properly.

chrome/browser/extensions/extension_settings_backend.h

Index: chrome/browser/extensions/extension_settings_backend.h
diff --git a/chrome/browser/extensions/extension_settings_backend.h b/chrome/browser/extensions/extension_settings_backend.h
index 43b90a3382657aaed07bc657ece3fcdc874c79f3..2ae8850fa676daeda4e12d8967b4bc245aa989a7 100644
--- a/chrome/browser/extensions/extension_settings_backend.h
+++ b/chrome/browser/extensions/extension_settings_backend.h
@@ -10,6 +10,7 @@
 #include "base/file_path.h"
 #include "base/memory/linked_ptr.h"
 #include "base/memory/ref_counted.h"
+#include "base/memory/weak_ptr.h"
 #include "base/task.h"
 #include "chrome/browser/extensions/syncable_extension_settings_storage.h"
 #include "chrome/browser/sync/api/syncable_service.h"
@@ -17,7 +18,9 @@
 // Manages ExtensionSettingsStorage objects for extensions, including routing
 // changes from sync to them.
 // Lives entirely on the FILE thread.
-class ExtensionSettingsBackend : public SyncableService {
+class ExtensionSettingsBackend
+    : public SyncableService,
+      public base::SupportsWeakPtr<ExtensionSettingsBackend> {
  public:
   // File path is the base of the extension settings directory.
   // The databases will be at base_path/extension_id.