On iPhone

After requesting the link, when using Mail or Gmail via Safari, after you long press on the sign-in link, you should be prompt with the following screen:

You should see the "open in Photoroom" option. If you tap on it, you should be redirected to the app.
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If the link is too old, then you might be prompted with an error message in Photoroom app. Don't worry, in that case, just resend a link to your email address and you should be fine.