Sending fit notes

I've been awarded ESA Support Group, letter dated 7th May 2019, do I need to continue sending fit notes, my last runs out on 1st June. Also the letter reads "This assessment is based on your National Insurance Contribution Record." My understanding is that because I am in the Support Group there is no time limit for ESA, none is mentioned in the letter ( I assume re-assessment will take place at the DWP's discretion). Can I confirm that this is the case for National Insurance Contribution ESA ?

Thank you Gordon, I appreciate both your clear answer to my questions and the time you take helping people like me through such a daunting claims procedure. I have a further question regarding my PIP claim, but I'm afraid it's very detailed and lengthy to explain the situation I find myself in, is this the correct format for seeking your advice or should I email my question and you can edit the relevant info to release and help others on the forum?