Gilbreath's conjecture is a hypothesis, or a conjecture, in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, Gilbreath observed a pattern while playing with the ordered sequence of prime numbers

2,3,5,7,11,13,17,19,23,29,31,... Computing the absolute value of the difference between the (n + 1)-th and n-th terms in this sequence yields the sequence

1,2,2,4,2,4,2,4,6,2,... If the same calculation is done for the terms in this new sequence, and the sequence that is the outcome of this process, and again ad infinitum for each sequence that is the output of such a calculation, the first five sequences in this list are given by

1,0,2,2,2,2,2,2,4,..., 1,2,0,0,0,0,0,2,..., 1,2,0,0,0,0,2,..., 1,2,0,0,0,2,..., and 1,2,0,0,2,...

Gilbreath claims that the first term in each series of differences appears to be 1.

I want to know recent research toward proving this conjecture.