WPF menu will not close when radiobutton group item is selected - wpf

Question

I created a menuitem resource style that assists in showing a group of mutually exclusive selectable radio button items. When one of the radio button items is selected the menu will not close. I tried using StaysOpenOnClick but that doesn't seem to work. How can I get the menu to close when one of the radio button menu items is selected?
XAML:
<Window x:Class="WpfApp1.MainWindow"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
xmlns:local="clr-namespace:WpfApp1"
mc:Ignorable="d"
Title="MainWindow" Height="200" Width="200">
<Grid>
<Menu>
<MenuItem Header="Menu" StaysOpenOnClick="False" >
<MenuItem.Resources>
<Style x:Key="GroupStyle1" TargetType="{x:Type RadioButton}">
<Setter Property="GroupName" Value="OptionGroup1"/>
</Style>
</MenuItem.Resources>
<MenuItem StaysOpenOnClick="False">
<MenuItem.Template>
<ControlTemplate>
<RadioButton
Content="Radio1"
Style="{StaticResource GroupStyle1}"/>
</ControlTemplate>
</MenuItem.Template>
</MenuItem>
<MenuItem StaysOpenOnClick="False">
<MenuItem.Template>
<ControlTemplate>
<RadioButton
Content="Radio2"
Style="{StaticResource GroupStyle1}"/>
</ControlTemplate>
</MenuItem.Template>
</MenuItem>
<MenuItem StaysOpenOnClick="False">
<MenuItem.Template>
<ControlTemplate>
<RadioButton
Content="Radio3"
Style="{StaticResource GroupStyle1}"/>
</ControlTemplate>
</MenuItem.Template>
</MenuItem>
</MenuItem>
</Menu>
</Grid>

Answer 1

I usually try not to use code behind with WPF, but for things like this, I think it may be justified. I'm sure if you think long enough you can write a whole bunch of xaml that can do this, but here's a quick and slightly dirty way of doing so:
Name the MenuItem
Add an event handler to the style, and use that to close the menu in the code behind. (See the EventSetter inside the style)
Xaml
<Grid>
<Menu>
<MenuItem x:Name="menuItem" Header="Menu">
<MenuItem.Resources>
<Style x:Key="GroupStyle1" TargetType="{x:Type RadioButton}">
<Setter Property="GroupName" Value="OptionGroup1"/>
<EventSetter Event="Checked" Handler="RadioButton_Checked"/> <!--Add this-->
</Style>
</MenuItem.Resources>
...
Code Behind
private void RadioButton_Checked(object sender, RoutedEventArgs e)
{
menuItem.IsSubmenuOpen = false;
}