Get a phone call with your connection of interest. Keep following up until the appointment happens. If you think email and SMS are going to close a deal you are wasting your time. Remember the old adage, most sales happen after the 5th Keep reaching out.
Move the conversation OUT of LinkedIn as fast as possible. Track people in a spreadsheet or your CRM system.
Nothing else matters except a phone appointment. You cannot sell effectively via chat. The LinkedIn mailbox is a horrible place to track conversations. You can grab people’s personal email addresses and sometimes their phone number from their contact info.