Conjecture on primitives of the greedy convolution of length 4

Let $M$ denote the primitives of rank $4$ for the greedy convolution of length $4$. The rank-one and rank-two primitives are the objects considered below, and divisibility and membership in $8M$ or $9M$ have their usual meanings. Conjecture on rank-one, rank-two, and rank-three primitives. The primitives of rank one and two are all divisible by $8$ or $9$; “most” primitives of rank one or two belong to $8M$ or $9M$; and there are no primitives of rank $3$. These claims are computationally motivated by the listed primitives, but the source gives no resolution of them.