When connecting Gorgias Chat to Chatdesk Teams, the Gorgias "email prompt" feature for chat must be set to "Always required". Here's Gorgias' explanation of the setting, from their documentation:
The email prompt feature will allow you to require your customers to enter their email before starting chat. This helps prevent tickets where your customers are leaving messages in the chat, but are then not reachable for a follow-up. It can be configured in the Preferences tab of your Chat settings.
There are two options for the email prompt:
Optional: the default option. Customers can send messages, but if they are not associated with an email address, they will be prompted to enter theirs.
Always required: customers cannot send a chat message without entering their email address first (except if their email address is already known).
Chatdesk on-demand Experts handle Gorgias Chat messages with the same SLAs as emails, and handle them in the same way. When they respond to Gorgias Chat messages, our platform uses the email address provided to Gorgias to ensure that the response reaches the end customer, even if they have left the chat.