Add gradle_desugar_test to the builders (#73901)

diff --git a/dev/prod_builders.json b/dev/prod_builders.json
index b546b6d..ffb3667 100644
--- a/dev/prod_builders.json
+++ b/dev/prod_builders.json
@@ -265,6 +265,12 @@
          "flaky": false
       },
       {
+        "name": "Linux gradle_desugar_classes_test",
+        "repo": "flutter",
+        "task_name": "linux_gradle_desugar_classes_test",
+        "flaky": false
+      },
+      {
          "name": "Linux gradle_non_android_plugin_test",
          "repo": "flutter",
          "task_name": "linux_gradle_non_android_plugin_test",
diff --git a/dev/try_builders.json b/dev/try_builders.json
index 6c961af..30f8d11 100644
--- a/dev/try_builders.json
+++ b/dev/try_builders.json
@@ -70,6 +70,13 @@
          "enabled":true
       },
       {
+        "name":"Linux gradle_desugar_classes_test",
+        "repo":"flutter",
+        "task_name":"linux_gradle_desugar_classes_test",
+        "enabled":true,
+        "run_if":["dev/**", "bin/**"]
+      },
+      {
          "name":"Linux gradle_non_android_plugin_test",
          "repo":"flutter",
          "task_name":"linux_gradle_non_android_plugin_test",