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.