Skype for Business compatibility with regular Skype

EDIT: I just found out the SFB in my org is on-prem. It may be related to the firewall. I will keep posting the updates here

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Our situation is worse than yours. The Skype user cannot initiate the conversation to the SFB users. For instance, when a Skype user sends a message to SBF user, the message is sent but never arrives SFB user. If SFB user sends a message first, the Skype user can receive and reply to the message. It's more strange that as soon as the SFB user closes chatbox, the issue comes back again that Skype user cannot send any message to SFB user. It feels like the session is closed.
If the Skype user calls SFB, it just pops up "Oops something went wrong". But the SFB user can initiate a call to Skype user. I tried with 4G and got the same result so the issue is not caused by the enterprise firewall.