When user clicks on Start a Conversation, the window is dismissed automatically.
Happens on my iPhone XR on iOS 15.6, but not on our test device which is on iOS 15.1.1.

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This doesn't seem like something that would have been caused by the connection being established to the conversation. Though I'd suggest opening up a support ticket so that we can see if your account had a disruption around the time you were experiencing this issue.
Thanks!