I apologize for the title. I myself conined it...

I am referring to **Conjecture 1.2 (page 7)** of Richard Taylor's paper Galois representations (Annales de la Faculte des Sciences de Toulouse 13 (2004) )
The pdf of the paper is here:

http://www.math.ias.edu/~rtaylor/longicm02.pdf

My question is:

What is the reference nowadays for this conjecture? What has been proved so far? I am particularly interested about **part 3**, i.e., the existence of a "common" set of Hodge-Tate weights.

And of course, I really would like to know the correct name of this conjecture...

Thanks!