I initially used Choosy because it supports multiple selected browsers, but it comes with a fee. Therefore, I switched to Velja.

However, Velja doesn't support opening an Inprivate tab of the Edge. So, I created an Automator and ShellScript to handle this issue.

You might encounter a dialog box prompting you to 'Copy open.sh' when selecting an incompatible browser.

This is the script.

#!/bin/sh
open "$@"

Therefore, I developed the following application in Automator. However, as discussed below, the script's content is meaningless.

Next, register the application you just created with Velja.

In addition, rewrite /Users/username/Library/Application Scripts/com.sindresorhus.Velja/open.sh as follows

#!/bin/sh
if [ "$2" == "com.apple.automator.Microsoft-Edge-Inprivate" ] 
then
	open -n -a "Microsoft Edge" --args --inprivate "$4"
else
	open "$@"
fi

This allows Velja to select the Inprivate tab of the Edge without problems.

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