I don't think that's been solved either or most physicists for that matter. I've read some of the attempts and they are not terrible good.
I don't think Ron Garret even touched on the Born rule, Everett of many worlds just hand waves and assumes it. The attempts I've seen are basically of the form that if the rule was different from the probability equaling the absolute amplitude squared then reality wouldn't come out as we find, so it must be so, which is not much of an explanation in my book.
It's true that I don't talk about the Born rule or multiple worlds in the paper, but I do talk about multiple worlds in the video (at the end, during the Q&A) and in the blog posts I linked to above.
As for the Born rule, what kind of an explanation are you hoping for? Some things can't be explained beyond, "That's just how it is." For example, why do the fundamental physical constants have the values that they have?
Two of the appealing arguments for many worlds type interpretations are that the number of required assumptions is small, and that it gives a reasonable physical intuition for why measurement outcomes occur as they do. Having to bolt on the Born rule weakens both of these.
In fact, I would even go so far as to say that if you are going to merely "enforce" the Born rule without justification, then you are no longer proposing an Everettian interpretation. The whole point of these interpretations is that simply "unitary quantum mechanics" describes the whole universe.
I agree that bolting on the Born rule seems a weakness of the many world view. It would be nice if you could say everything follows the Schrodinger equation and hence the probabilities naturally come out the way they do but it doesn't seem to work that way very easily.
Everett himself however seems to basically "enforce" the Born rule without justification so I guess that would be Everettian.
Probability is a funny thing to deal with. Say you have an experiment where you push a button and a red light or green comes on based on some quantum effect but the green light is 1000x more likely. In many worlds you'd end up with different worlds with an observer seeing one or another but it's tricky to see how the 1000x thing comes in.
I don't think Ron Garret even touched on the Born rule, Everett of many worlds just hand waves and assumes it. The attempts I've seen are basically of the form that if the rule was different from the probability equaling the absolute amplitude squared then reality wouldn't come out as we find, so it must be so, which is not much of an explanation in my book.