To check the chat you have with the customer, you can check the SMS logs that you have already sent. For this:
Step 1: From the phone setting page, select the logs tab.
Step 2: You can filter the logs on the basis of the columns in the logs and save the most used filter as well for future use.
For this select Add New Filter.
Date & Time - indicates the date and time the SMS was sent.
From Number - Indicates the number from which the SMS was sent.
To Number - Indicates to which the SMS was sent.
Type - Indicates the type of SMS that is Inbox or Outbox.
Status - Indicates the status of the SMS that is Queued, Sent, Delivered or Undelivered.
Queued means the SMS is not sent but it is queued to be sent.
Sent means the phone carrier have sent the text message.
Failed means the message have not been delivered. Check if its right phone number if this status appears.
Delivered means the SMS has successfully been delivered.
Undelivered means the SMS has not delivered yet. If the SMS is undelivered please confirm with the number.