notify network on Linkedin?

So I recently got a new job. It's my dream job. The thing is I bent the truth to get it. I said I was laid off from previous employer where in fact I was let go despite hitting targets.. I didn't play the long corporate game I should have. Fully my fault although not fully within my control..

I know I know.. lying should be avoided but being honest just didn't get me to next stage on interviews.. and it had been almost 9 months!

the question I have is in your opinion should I 'notify feed' on LinkedIn when posting about the new or do you think I should keep on the d-low?

I'm afraid that if I notify network someone may reach out to my new employer and jeapordise the opportunity!

Thanks