A random analogue of Gilbreath's conjecture

A well-known conjecture of Gilbreath, and independently Proth from the 1800s, states that if $a_{0,n} = p_n$ denotes the $n^{\text{th}}$ prime number and $a_{i,n} = |a_{i-1,n}-a_{i-1,n+1}|$ for $i, n \ge 1$, then $a_{i,1} = 1$ for all $i \ge 1$. It has been postulated repeatedly that the property of having $a_{i,1} = 1$ for $i$ large enough should hold for any choice of initial $(a_{0,n})_{n \ge 1}$ provided that the gaps $a_{0,n+1}-a_{0,n}$ are not too large and are sufficiently random. We prove (a precise form of) this postulate.