Finite-rank conjecture for nilpotent locally compact p-groups

Let $G$ be a nilpotent locally compact $p$-group, and let $\mathrm{rank}_p(G)$ denote its $p$-rank. The finite-rank conjecture. If

$$\mathrm{rank}_p(G)<\infty,$$

then $G\in\mathfrak E_{<\infty}$, meaning that every continuous endomorphism of $G$ has finite topological entropy. The conjecture is known for topological automorphisms, by the Addition Theorem for topological automorphisms of totally disconnected locally compact groups, but remains open for arbitrary continuous endomorphisms.