There is now some slight theoretical progress towards the conjecture in

Chase, Zachary, A random analogue of Gilbreath’s conjecture, ZBL07808058.

If one models the prime gaps $p_{n+1}-p_n$ (beyond the first gap $p_2-p_1=1$) as an even number between $2$ and $2f(n)$ chosen uniformly and independently at random for some slowly growing function, then in this paper the analogue of Gilbreath's conjecture is established almost surely for sufficiently large $n$ provided that $f(n)$ grows slower than $\frac{1}{100} \frac{\log\log n}{\log\log\log n}$. For comparison, the Cramér random model roughly corresponds to taking $f(n) = \log n$ (and also replacing the uniform distribution by an exponential distribution of the same mean). So this is not yet a fully satisfactory heuristic justification towards the conjecture (on the level of, say, the heuristic justification of the twin prime conjecture based on Cramér type models), but is at least a good first step in that direction.

This paper also dug into the historical claim that Proth had claimed a proof of this conjecture, but it appears that this claim was based on a misreading of the text and has since been retracted by its originator.

There is some theoretical progress towards the conjecture in

Chase, Zachary, A random analogue of Gilbreath’s conjecture, ZBL07808058.

If one models the prime gaps $p_{n+1}-p_n$ (beyond the first gap $p_2-p_1=1$) as an even number between $2$ and $2f(n)$ chosen uniformly and independently at random for some slowly growing function, then in this paper the analogue of Gilbreath's conjecture is established almost surely for sufficiently large $n$ provided that $f(n)$ grows slower than $\frac{1}{100} \frac{\log\log n}{\log\log\log n}$. For comparison, the Cramér random model roughly corresponds to taking $f(n) = \log n$ (and also replacing the uniform distribution by an exponential distribution of the same mean). So this is not yet a fully satisfactory heuristic justification towards the conjecture (on the level of, say, the heuristic justification of the twin prime conjecture based on Cramér type models), but is at least a good first step in that direction.

This paper also dug into the assertion that Proth had claimed a proof of this conjecture, but it appears that this assertion was based on a misreading of the text and has since been retracted by its originator.

EDIT: I guess this would be a good place to mention some of the ideas of the proof. Consider a large block of random gaps and take successive absolute value differences. If one can get these differences down to 2 or less then one is done, and the maximum size of the differences is non-increasing as one moves from one row to the next, so the only way one can get "stuck" is if the maximum difference stays at some level $2d>2$ for a very long time. This turns out to mean that some rows contain very long blocks that consist only of $0$ and $2d$. This is easy to rule out with high probability if $d$ is even, as successive differences modulo 4 are tractable to compute. For odd $d$ one has to work harder. After using a Cauchy-Schwarz argument to make these blocks slightly longer (with some positive probability), and again using the control of differences mod 4, the author then shows that the $0$ value must occur with some reasonably large frequency at every row. This turns out to make it very unlikely for long blocks of $0$ and $2d$ to form (as the row just before the first appearance of such a block would create a long block without any $0$s whatsoever).

There is now some slight theoretical progress towards the conjecture in

Chase, Zachary, A random analogue of Gilbreath’s conjecture, ZBL07808058.

If one models the prime gaps $p_{n+1}-p_n$ (beyond the first gap $p_2-p_1=1$) as a random even number between $2$ and $2f(n)$ for some slowly growing function, then in this paper the analogue of Gilbreath's conjecture is established almost surely for sufficiently large $n$ provided that $f(n)$ grows slower than $\frac{1}{100} \frac{\log\log n}{\log\log\log n}$. For comparison, the Cramer random model roughly corresponds to taking $f(n) = \log n$. So this is not yet a fully satisfactory heuristic justification towards the conjecture, but is at least a good first step in that direction.

There is now some slight theoretical progress towards the conjecture in

Chase, Zachary, A random analogue of Gilbreath’s conjecture, ZBL07808058.

If one models the prime gaps $p_{n+1}-p_n$ (beyond the first gap $p_2-p_1=1$) as an even number between $2$ and $2f(n)$ chosen uniformly and independently at random for some slowly growing function, then in this paper the analogue of Gilbreath's conjecture is established almost surely for sufficiently large $n$ provided that $f(n)$ grows slower than $\frac{1}{100} \frac{\log\log n}{\log\log\log n}$. For comparison, the Cramér random model roughly corresponds to taking $f(n) = \log n$ (and also replacing the uniform distribution by an exponential distribution of the same mean). So this is not yet a fully satisfactory heuristic justification towards the conjecture (on the level of, say, the heuristic justification of the twin prime conjecture based on Cramér type models), but is at least a good first step in that direction.

This paper also dug into the historical claim that Proth had claimed a proof of this conjecture, but it appears that this claim was based on a misreading of the text and has since been retracted by its originator.