Open problems in Banach spaces? universal spaces

Question

I have gathered a list of universality problems in Banach spaces which have been solved:

1. The non existence of a separable reflexive space universal for the class of separable reflexive spaces.
2. If a space is universal for the class of separable reflexive spaces, then it is universal for the class of separable Banach spaces.
3. There is a separable reflexive space which is universal for all separable uniformly convex spaces.

My question is, are there any open problems in a similar vein to these?