Reidemeister classes in some weakly branch groups

We prove that a saturated weakly branch group $G$ has the property $R_{\mathrm{\infty }}$ (any automorphism $\varphi :G\to G$ has infinite Reidemeister number) in each of the following cases:

1. any element of $Out(G)$ has finite order;
2. for any $\varphi$ the number of orbits on levels of the tree automorphism $t$ inducing $\varphi$ is uniformly bounded and $G$ is weakly stabilizer transitive;
3. $G$ is finitely generated, prime-branching, and weakly stabilizer transitive with some non-abelian stabilizers (with no restrictions on automorphisms). Some related facts and generalizations are proved.