Yes, for lead connect to work you need to make sure your browser is open for at least 4-5 hours a day. (for 100 invitations and followups).
As we do not take any of your LinkedIn login details, our tool only work when you have logged in to LinkedIn. (No need to keep the tab open but make sure you don't logout)
We do not send any request from our servers so that all activity seems to be coming from your browser only and for this we need a browser to be kept open.