DataTrigger inside ControlTemplate doesn't update - wpf

Question

I have a ListBox that is bound to a list of CustomerViewModel-objects, that each has two dependency properties:
- Name (string)
- Description (string)
- IsVisible (bool)
(the IsVisible property is True by default and is reversed via the ToggleVisibility Command on the CustomerViewModel)
I would like to display the Name and Description to the right of a Border-control, that is has a Transparent background when the IsVisible property is True and Green when the False.
My problem is that the DataTrigger part of the code below doesn't work the way I want, because the Setter-part isn't triggered when the IsVisible is changed.
What am I doing wrong?
Here's my code:
<UserControl.Resources>
<Style x:Key="ListBoxStyle" TargetType="{x:Type ListBox}">
<Setter Property="Margin" Value="-1,-1,0,0" />
<Setter Property="BorderThickness" Value="0" />
<Setter Property="Background" Value="Transparent" />
<Setter Property="ItemContainerStyle" Value="{DynamicResource ListboxItemStyle}" />
<Setter Property="ScrollViewer.HorizontalScrollBarVisibility" Value="Disabled" />
</Style>
<Style x:Key="ListboxItemStyle" TargetType="{x:Type ListBoxItem}">
<Setter Property="Background" Value="Transparent" />
<Setter Property="FocusVisualStyle" Value="{x:Null}" />
<Setter Property="Template">
<Setter.Value>
<ControlTemplate TargetType="{x:Type ListBoxItem}">
<Grid>
<Border x:Name="border" Background="{TemplateBinding Background}" BorderBrush="#FFD4D6D5" BorderThickness="0,0,0,1">
<Grid Height="70" Margin="0,0,10,0">
<Grid.RowDefinitions>
<RowDefinition Height="10" />
<RowDefinition Height="Auto" />
<RowDefinition />
<RowDefinition Height="10" />
</Grid.RowDefinitions>
<Grid.ColumnDefinitions>
<ColumnDefinition Width="Auto" />
<ColumnDefinition />
</Grid.ColumnDefinitions>
<Border x:Name="visibilityColumn" Grid.Row="0" Grid.Column="0" Grid.RowSpan="4" Background="Transparent" Width="4" Margin="0,0,4,0" />
<TextBlock x:Name="customerName" Grid.Row="1" Grid.Column="1" Foreground="#FF191919" FontWeight="Bold" Text="{Binding Name}" VerticalAlignment="Top" />
<TextBlock Grid.Row="2" Grid.Column="1" VerticalAlignment="Stretch" Text="{Binding Description}" TextWrapping="Wrap" Foreground="#FFB4B4B4" TextTrimming="CharacterEllipsis" />
</Grid>
<Border.ContextMenu>
<ContextMenu>
<MenuItem Header="Edit..." />
<MenuItem Header="Visible" IsCheckable="True" IsChecked="{Binding IsVisible}" Command="{Binding ToggleVisibility}"/>
</ContextMenu>
</Border.ContextMenu>
</Border>
</Grid>
<ControlTemplate.Triggers>
<Trigger Property="IsMouseOver" Value="True">
<Setter Property="Background" Value="#FFEEEEEE" />
</Trigger>
<Trigger Property="IsSelected" Value="True">
<Setter Property="Background" Value="#FFF5F5F5" />
<Setter TargetName="customerName" Property="Foreground" Value="Green" />
</Trigger>
<DataTrigger Binding="{Binding IsVisible}" Value="False"> <!--If Value="True" the customerName Border shows up green!-->
<Setter Property="Background" Value="Green" />
</DataTrigger>
</ControlTemplate.Triggers>
</ControlTemplate>
</Setter.Value>
</Setter>
</Style>
</UserControl.Resources>
<ListBox Style="{StaticResource ListBoxStyle}" ItemsSource="{Binding CustomerViewModels}" />
UPDATED:
The DataTrigger was indeed missing the TargetName="visibilityColumn" as pointed out by Goblin.
However - the "real" problem was, this line:
<MenuItem Header="Visible" IsCheckable="True" IsChecked="{Binding IsVisible}" Command="{Binding ToggleVisibility}"/>
A checkable MenuItem has a TwoWay binding-mode on the IsChecked-property, so I was actually inverting the IsVisiblity twice - via databinding and via the ToggleVisibility command... Whoops :)

Answer 1

Try switching this part:
<DataTrigger Binding="{Binding IsVisible}" Value="False">
<Setter Property="Background" Value="Green" />
</DataTrigger>
With this part:
<DataTrigger Binding="{Binding IsVisible}" Value="False">
<Setter TargetName="visibilityColumn" Property="Background" Value="Green" />
</DataTrigger>
I think you missed the TargetName property in your setter. (Same goes for your IsSelected- and IsMouseOver-trigger by the way)
Hope this helps!