Justin, As I mentioned it is not going to be possible for you then. It would require custom code in the WidgetManager of the viewer or the HeaderControllerWidget. You will have to settle for educating your users on the need to close one before opening another (maybe put that in the splash page).
Justin, That would only be possible by editing the code for the viewer do you have Flash Builder and the Flex Viewer source code?
<widget label="Louisville Police (QueryWidget)" width="300" height="600" icon="assets/images/i_policestation.png" config="widgets/Query/QueryWidget_Louisville_PoliceStations.xml" url="widgets/Query/QueryWidget.swf"/>
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