Finite Gilbreath conjecture for prime circuits

Let $\mathbb{P}$ denote the set of all prime numbers. For $n\geq 2$, let $\{p_i\}_{i=1}^n$ be an originator consisting of prime numbers, and let $\{d_j^k\}_{j\geq 1}$, for $1\leq k\leq n-1$, be the circuit induced by this originator. Here $d_j^k$ denotes the entries obtained by iterated absolute forward differences, and $\tau_{n,1}$ denotes the trace statistic of the circuit.

Finite Gilbreath conjecture. For every such prime originator,

$$d_1^k>0\qquad\text{for all }1\leq k\leq n-1,$$

and

$$\tau_{n,1}=n-1\qquad\text{for all }n\geq 2.$$

This is presented as a finite formulation whose establishment would immediately imply the original Gilbreath conjecture. The source introduces it in terms of paths, circuits, traces, and lengths; the supplied context does not provide their full formal definitions, so the precise relationship between the trace condition and the original conjecture should be checked.