Am i eligible for a nenkin exemption?

Hi everyone,
​I’m seeking some advice regarding National Pension (Kokumin Nenkin) payments. Here is my situation:
​Arrival: I moved to Japan in August 2025 for a new job.
​Income: I currently earn 200k yen/month, but my 2024 income in Japan was zero (I wasn't in the country yet).
​Status: I am currently 4 months late on my payments (August, September, October, November). I just received a large envelope with a bunch of slips, including a 6-month prepayment slip for ~104k yen.
​Visa: I’m on a work visa and worried that being "late" will hurt my future renewals.
​My questions:
​Am I eligible for the Full Exemption (Menjo) for these months since my 2024 Japanese income was zero?
​Does my current salary (200k/mo) disqualify me, or is it strictly based on last year's Japanese tax records?
​If I apply for the exemption now at City Hall, will the "late" status for the past 4 months be cleared/legally covered? I don't want to pay the 104k yen bulk bill if I'm legally entitled to an exemption.
​How does an approved exemption look to Immigration compared to just paying the bills late?