The non-overlapping solitons conjecture

Let $p$ be a prime and let $g_1,g_2\geq 2$. For each $g$, let $\mathfrak S(p,g)$ be the connected collection of cells that starts at $p^g$ and contains only powers of $p$ larger than two. These collections are called $Z$-solitons. Non-overlapping solitons conjecture. Two distinct solitons $\mathfrak S(p,g_1)$ and $\mathfrak S(p,g_2)$ neither overlap nor touch each other. The conjecture is verified for $p=2$ by the complete description of the solitons in that case; it remains open for odd primes, whose solitons have a more complicated structure.