Triggers override binding - wpf

Question

Why is it that the binding Marked stops working when the triggers are used? Is there some way to fix this?
The multi selection ListBox has ListBoxItems with CheckBoxes, and the checkbox in an item is ticked when the corresponding item is selected, using mouse or keyboard or touch or whatever. The point with this demo is to not tick the checkboxes directly, but just multiselect the items and see the checkboxes get ticked as a consequence of that. This synchronization in the XAML works well, but since the Marked binding isn't working, the model won't get updated.
If I remove the triggers, then the binding Marked starts working. I know, because then the model gets updated when the checkboxes are ticked.
If I attach event handlers for Checked and Unchecked to the CheckBox, they fire even when the triggers are there.
<ListBox ItemsSource="{Binding Lines}" SelectionMode="Extended">
<ListBox.ItemTemplate>
<DataTemplate>
<StackPanel>
<CheckBox x:Name="cb"
Content="{Binding DisplayText}"
IsChecked="{Binding Marked, Mode=TwoWay}">
</CheckBox>
</StackPanel>
<DataTemplate.Triggers>
<DataTrigger Binding="{Binding RelativeSource={RelativeSource FindAncestor,
AncestorType={x:Type ListBoxItem}}, Path=IsSelected}" Value="true">
<Setter TargetName="cb" Property="IsChecked" Value="true"/>
</DataTrigger>
<DataTrigger Binding="{Binding RelativeSource={RelativeSource FindAncestor,
AncestorType={x:Type ListBoxItem}}, Path=IsSelected}" Value="false">
<Setter TargetName="cb" Property="IsChecked" Value="false"/>
</DataTrigger>
</DataTemplate.Triggers>
</DataTemplate>
</ListBox.ItemTemplate>
</ListBox>
EDIT:
Although the accepted solution below is good XAML, it crashes with Elmish.WPF, which I'm using. That's another issue, and will be solved through other forums.

Answer 1

The DataTriggers will obviously override the Binding, since the ListBoxItem's IsSelected property is always either true or false.
Replace the DataTriggers by an ItemContainerStyle:
<ListBox ItemsSource="{Binding Lines}" SelectionMode="Extended">
<ListBox.ItemTemplate>
<DataTemplate>
<CheckBox Content="{Binding DisplayText}"
IsChecked="{Binding Marked}"/>
</DataTemplate>
</ListBox.ItemTemplate>
<ListBox.ItemContainerStyle>
<Style TargetType="ListBoxItem">
<Setter Property="IsSelected" Value="{Binding Marked}"/>
</Style>
</ListBox.ItemContainerStyle>
</ListBox>