That's an absolute cause for concern. 4% is already pretty slim, fluff could bring it to 0% rather quickly. 

Seem like profit margins are really slim.

I agree with all the above that the odds are slim, however, check with your local in-the-know contacts/lawyers/investors. 

Really pushing for a slim budget - will do whatever I can myself!!

The area is great and appreciating at an unprecedented rate so even if I come out of pocket 10k after the refi I’d be fine it’s just the cash flow numbers that start getting slim with the rates that I’m the most concerned about.

If you have 10 or more the chances of having enough units empty long enough not to be able to cover payments is very slim.

This will slim your list to make it less daunting.Lease Agreements:  You have to honor the previous lease agreements, unless they're month-to-month.  

Chances of finding something sitting in MLS that's going to cash flow with 20 or 25% down is slim in most areas I know of.

I'm in the south shore area where there are very slim pickings.

It's not the 1960s anymore and the odds of the OP having issues because of her name are slim and none (and Slim just left town).