Caterpillar classification conjecture for Penrose caterpillars

Let $C$ be a Penrose caterpillar, let $S$ be the graded poset of caterpillars used in the source, and let $f_{boat}$ transform a Penrose caterpillar by replacing each relevant subpath along the hull of a boat with the corresponding subpath along the top of the boat. Write $(f_{boat}(C))'$ for its derived path. Caterpillar classification conjecture. $C$ belongs to one of the following two sets:

(i) $C$ belongs to the graded poset $S$.

(ii) There exists a caterpillar $C_k$ such that the derived path $(f_{boat}(C))'$ is included in the derived path $(f_{boat}(C_k))'$, up to isomorphism.

The conjecture is intended to describe the possible shapes of fully leafed Penrose trees through the family of Penrose caterpillars and the boat-path transformation. The source states no result resolving this classification.