Can I ask for shorter notice period than specified in non-compete agreement if not competing?

I was asked to sign a non-compete agreement with my current employer. It requires 3 months notice period with possibly another 3 months before I can work for an competitor. The scope is related to any work that I did in the past year. It also says at firm's discretion, it can shorten the period as appropriate. I just found a job that's in the same industry, but not in competition with my employer. The role also has nothing to do with what I am doing right now. The job function is entirely different. I have followed the agreement to tell my employer which firm I will be going to at time of resignation. Given the fact that my new employer is not competing with my current employer, can I ask my employer to shorten their non-compete notice period and coverage period requirement? If they do not give me a regular notice period and coverage period, which may impact my new job because it's close to end of year, can I sue them?