Is the 3x income requirement for rentals strictly followed?

I am planning to apply for residency in Leiden under the DAFT pathway and meet all requirements, however my current semi-retirement monthly income is about 3400 Euro. I do have considerable savings and strong credit ratings, and I am comfortable spending about 2000 Euro/month. From scanning ad postings I think the 3x standard would rule out any place in Leiden I would want to live (1+ bedrooms). Anyone have insight that would help me decide whether this is a lost cause? It's quite possible my income would increase somewhat over time in spurts with contracts, but not certain and not provable.
thanks in advance! - @kgeographer

Yes, it does tend to be a strict rule that landlords follow, but saying that, if you find a private landlord, you can offer a larger deposit, and you are successful, you should be alright.

Apologies, only just picked up on this.  I totally agree with Simons (SimCity AT) remarks above; just to add, many landlords try to say this is Dutch law, it is not, it's a self-imposed guideline, many say it's because it is so difficult to remove a tenant if they fail to pay their rent; whatever, Simons advice is sound.

Appreciate the candid replies SimCity and Cynic. Unfortunately this makes it a low odds proposition for me, and probably unwise to do all the DAFT prep and arrive on the chance I can find housing that works. Escape paths from the US are narrowing. Ah well. I can stay and "resist" or escape to Thailand I guess.