Chat message sent under SR > Messages tab not sent to contact

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I have implemented all the Chat Setup steps on OEC using the Chat Inlay option and the Chat communications and chat assignment rules are working perfectly and the chat interactions and chat queue assignment are working perfectly. 

During a chat interaction after created an SR associated to a contact I navigate under the Messages tab for the associated SR and configure a Response message by selecting the Chat Channel. 

After that I click on Send button but the contact does not receive this chat message.

I suppose the chat message should be received by the contact on the contact Chat Inlay screen during the chat interaction.