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Test with latest Xamarin.Forms(5.0.0.2012) , it works without problem , the Flyout menu does not close when tapping on the expander .
Code
<Shell.FlyoutHeader>
<xct:Expander>
<xct:Expander.Header>
<Label Text="Click to Expand" FontAttributes="Bold" FontSize="Medium" />
</xct:Expander.Header>
<Grid Padding="10">
<Grid.ColumnDefinitions>
<ColumnDefinition Width="Auto" />
<ColumnDefinition Width="Auto" />
</Grid.ColumnDefinitions>
<Image Source="xamarinstore.jpg"
Aspect="AspectFill"
HeightRequest="120"
WidthRequest="120" />
<Label Grid.Column="1"
Text="Hello I'm Cole"
FontAttributes="Italic" />
</Grid>
</xct:Expander>
</Shell.FlyoutHeader>
Screen shot
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