The multiplicative Gilbreath conjecture for the left edge

Let $T_{\mathbb N^*}$ be the infinite triangle generated by iterating the $Z$-rule on the positive integers, and call its first column the left edge. A number is square-free when no square of a prime divides it. Multiplicative Gilbreath conjecture. Every number on the left edge of $T_{\mathbb N^*}$ is square-free; equivalently, the maximal power of any prime occurring in the prime factorization of a left-edge entry is one. This is the multiplicative analogue of the Gilbreath conjecture and is supported by computer evidence; the source notes that it is proved for the solitons $\mathfrak S(2,g)$, but does not establish the assertion for the entire left edge.