Extendability conjecture for sea grafting (g) in Penrose P2 tilings

Consider the sea grafting (g) shown in Figure of the paper. A bi-infinite fully leafed caterpillar is a fully leafed caterpillar that extends infinitely in both directions. Extendability conjecture. The sea grafting (g) can be extended into a bi-infinite fully leafed caterpillar. The paper presents this as an unresolved construction problem; the authors expect that proving it will require a non-trivial construction related to their second bi-infinite caterpillar.