To ensure Instagram DM access for third-party apps such as Chatdesk, please follow the steps below. More information about 3rd party application setting for Instagram can be found here.
Open the Instagram App on your phone & switch to your business account
Tap on the profile icon in the bottom right corner & then Tap on the settings menu (three lines) in the top right
Now tap Settings & navigate to Privacy > Messages
Under the 'Connected Tools' heading, turn on Allow Access to Messages