How can I avoid the entire HTML code that comes out when the comment section is included in the message sent to Slack?

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When you are using dynamic variables to configure custom messages for Slack, please use the following variables for ticket description, last public note, and last private note so that only the text content of the description and notes is sent to Slack. 

Otherwise, the HTML tags will get pushed to slack as well.

You can replace the placeholder with {{ticket.description_text}} or {{ticket.description | strip_html}} instead of {{ticket.description}} and this will just have the text portion included in the notification.