The Addition Theorem conjecture for locally finite groups

Let $G$ be a group. For every endomorphism $\phi\in\operatorname{End}(G)$ and every $\phi$-invariant normal subgroup $H$ of $G$, write $AT(G,\phi,H)$ when

$$h(\phi)=h(\phi\mathbin\restriction_H)+h(\bar\phi_{G/H}),$$

where $\bar\phi_{G/H}\in\operatorname{End}(G/H)$ is induced by $\phi$. The Addition Theorem holds for $G$, written $AT(G)$, if $AT(G,\phi,H)$ holds for every such $\phi$ and $H$.

Locally finite-group Addition Theorem conjecture. If $G$ is a locally finite group, then $AT(G)$ holds.

The conjecture was stated in earlier work and is the principal question addressed by the paper. The source indicates that it is proved in a particular case, so the general assertion remains open.