To publish director as an application and run it in kiosk mode, below steps can be followed:
 

1. Publish EDGE as an application in Citrix Studio.
2. Go to the properties of published application and on the location tab, enter the director URL in the command line argument as follows:  --kiosk  http://<<DDC FQDN or IP address >>/Director/LogOn.aspx?ReturnUrl=%2fdirector&cc=true
3. Post that test and verify that you are able to launch the director app from store

Please check the Microsoft documentation to learn more about the kiosk mode.

Note: Kiosk mode helps you create a dedicated and locked down user experience.
 

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The above steps can be used as a workaround to publish director as an application for the users