Caterpillar decomposition conjecture for fully leafed Penrose subtrees

Let $T$ be a fully leafed Penrose subtree. A Penrose tree is saturated if $n_{1}(T)=\overline{L_{P2}}(n(T))$, where $\overline{L_{P2}}(n)=8n/17+b$ is the unique linear upper bound described in the source, and an appendix is a fully leafed induced subtree whose derived subtree contains precisely one leaf of $T$-degree $2$ and at most two cells of $T$-degree $3$. Caterpillar decomposition conjecture.

(i) If $T$ is saturated, then $T$ is a caterpillar.

(ii) If $T$ is non-saturated, then there exist a saturated caterpillar $T_1$ and an appendix $T_2$ such that

$$T=T_1\diamond T_2.$$

This conjecture proposes that saturated fully leafed Penrose subtrees are exactly caterpillars, while every non-saturated one is obtained by grafting an appendix onto a saturated caterpillar. The source gives this as a formalization of observed structures; no resolution is stated.