How conditional are conditional offers
An awfully far worse way to start the relationship would be to not be able to start that relationship. Please do yourself a favor and bite the bullet, contact the relevant person right away and calmly explain your current lapse/fear -  then listen carefully to what they tell you. There is no standard or law on how to handle such a situation, the program is likely able to do pretty much whatever it wants. The sooner you find something out the more time you have to work with to try to work something out - or to at least be able to relax!

I'd go so far as to say that if someone outside your program tells you with any certainty whether or not such an offer will be rescinded or not, you really should ignore them utterly - talk with your program! Every University, department, and program is pretty unique unto itself in questions of graduate school admission.

This is just one of those things best handled directly with the responsible persons, as advice in this form is worth it's mass in gold. Good luck - deep breaths!