Switch BookmarkFaviconFetcher to the TaskScheduler API.

chrome/browser/bookmarks/bookmark_html_writer.cc

Index: chrome/browser/bookmarks/bookmark_html_writer.cc
diff --git a/chrome/browser/bookmarks/bookmark_html_writer.cc b/chrome/browser/bookmarks/bookmark_html_writer.cc
index 2ebc2621949aa5cae8166b75084bacc2d51b5bc1..7583c88748f5123f74d8d31ea3102da9a9253241 100644
--- a/chrome/browser/bookmarks/bookmark_html_writer.cc
+++ b/chrome/browser/bookmarks/bookmark_html_writer.cc
@@ -448,8 +448,9 @@ void BookmarkFaviconFetcher::ExecuteWriter() {
   // for the duration of the write), as such we make a copy of the
   // BookmarkModel using BookmarkCodec then write from that.
   BookmarkCodec codec;
-  BrowserThread::PostTask(
-      BrowserThread::FILE, FROM_HERE,
+
+  background_io_task_runner_->PostTask(
+      FROM_HERE,
       base::BindOnce(
           &Writer::DoWrite,
           new Writer(codec.Encode(