Equivalence conjecture for amoeba immortality notions

Let $A$=(H,m) be an amoeba. An amoeba is immortal if it admits an infinite growth sequence, strongly immortal if every relevant growth process can continue indefinitely, and partially immortal if there is an infinite growth sequence from some starting configuration.

Immortality-equivalence conjecture. The following properties of an amoeba $A$ are equivalent:

• $A$ is immortal.
• $A$ is strongly immortal.
• $A$ is partially immortal.

In general, strong immortality implies immortality, while the converse is open; for extension parameters $\ell\in\{1,2\}$ the converse is known. The conjecture would identify the three notions of immortality, leaving no distinction between them.