The un-replied conversations icon will display all incoming text messages, emails and chat messages from your customer's that are awaiting a response.
Responding to your customers in a timely manner can have a major impact on customer engagement. You will notice a running timer next to each customer awaiting response. This timer will continue to run until a response has been sent to the customer message and will impact response time reporting metics.
NOTE: An ignore option will appear when hover your cursor over an un-replied message. Best practice, is to always have the last response when communicating to a customer as this can also impact response time reporting metrics.