After I inserted an OverlayEntry, I would like to disable any user interaction on the underlying Widget so that you can only interact with the OverlayEntry. How could I do this?

I’ve run the example code below inside Chrome. After I’ve clicked on the button to open the OverlayEntry, I can just enter TAB which focuses the TextField where I can enter text.

Is it possible to make the OverlayEntry absorb any key input so that it doesn’t reach the Widget underneath it? Or failing that, is it somehow possible to fully disable the Widget’s user interaction until the OverlayEntry is closed?

import 'package:flutter/material.dart';

void main() {
  runApp(const MyApp());
}

class MyApp extends StatelessWidget {
  const MyApp({super.key});

  // This widget is the root of your application.
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Demo',
      theme: ThemeData(
        colorScheme: ColorScheme.fromSeed(seedColor: Colors.deepPurple),
        useMaterial3: true,
      ),
      home: const MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  const MyHomePage({super.key, required this.title});

  final String title;

  @override
  State<MyHomePage> createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> {
  OverlayEntry? overlayEntry;

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      appBar: AppBar(
        backgroundColor: Theme.of(context).colorScheme.inversePrimary,
        title: Text(widget.title),
      ),
      body: Center(
        child: Column(
          mainAxisAlignment: MainAxisAlignment.center,
          children: <Widget>[
            const TextField(
              decoration: InputDecoration(
                labelText: "Input",
              ),
            ),
            TextButton(
              onPressed: () {
                overlayEntry = _newOverlayEntry();
                Overlay.of(context).insert(overlayEntry!);
              },
              child: const Text("Open Overlay"),
            ),
          ],
        ),
      ),
    );
  }

  OverlayEntry _newOverlayEntry() {
    return OverlayEntry(
      builder: (context) {
        final Size mediaSize = MediaQuery.of(context).size;

        return Container(
          color: Theme.of(context).disabledColor,
          width: mediaSize.width,
          height: mediaSize.height,
          child: Align(
            alignment: Alignment.centerRight,
            child: Container(
              width: 200,
              height: 200,
              color: Colors.green,
              child: TextButton(
                onPressed: () {
                  overlayEntry?.remove();
                  overlayEntry?.dispose();
                  overlayEntry = null;
                },
                child: const Text("Close"),
              ),
            ),
          ),
        );
      },
    );
  }
}