I don't think that there has been any progress on this conjecture except for numeric verification (see Odlyzko, A. M. "Iterated Absolute Values of Differences of Consecutive Primes"), and if you search for publications with the appropriate key words you won't find any more papers after that. It just seems that it is one of those conjectures that are easy to come up with but that are too distant from the rest of mathematics to have a decent chance to be solved in the near future. In <a href="http://www.johndcook.com/blog/2009/09/09/gilbreath-conjecture/">this</a> blog post I found the following amusing line
> Paul Erdős speculated that Gilbreath’s conjecture is true but it would be 200 years before anyone could prove it. I find Erdős’s conjecture more interesting than Gilbreath’s conjecture.

Another conjecture that would possibly shed some light on this is that the result isn't actually related to prime numbers but it holds for any sequence of appropriate growth rate, but I don't know what the opinion of the specialists is.