Infinite algebraic entropy conjecture for automorphisms of exponentially growing groups

Let $G$ be a finitely generated group such that $h_{alg}(id_G)=\infty$, equivalently, $G$ has exponential growth. Infinite algebraic entropy conjecture. Then $h_{alg}(\phi)=\infty$ for every automorphism $\phi:G\to G$. A weaker form asks whether $h_{alg}(\phi)=\infty$ for every internal automorphism $\phi:G\to G$, that is, every automorphism given by conjugation by an element of $G$. The proposition immediately preceding this formulation establishes the corresponding statement for the identity endomorphism and characterizes exponential growth by $h_{alg}(id_G)=\infty$; the automorphism and internal-automorphism assertions are posed as further questions.