Gilbreath's conjecture

Gilbreaths conjecture is an unproven number-theoretic statement, which is attributed to the 1958 Norman L. Gilbreath (* 1936) and was published in 1878 by François Proth together with a later proven to be erroneous evidence. It relates to the primes. Gilbreath is said to have discovered while doodling on a napkin.

Should be written in a first line of the sequence of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ... Then, calculate the absolute value of difference between two consecutive sequence elements and record as the second line. Just fancy you the third and all following lines:

Gilbreaths guess is now that the first value is of each line except the first line 1. Andrew Odlyzko gave a check for approximately the first row.