The twisted conjugacy separability domination conjecture

Let $N$ be a nilpotent group with generating subset $S$. For an automorphism $\varphi\in\operatorname{Aut}(N)$, let $\operatorname{Conj}_{N,S}^{\varphi}(n)$ denote the effective twisted conjugacy separability function, and let $\operatorname{Conj}_{N,S}(n)$ denote the ordinary conjugacy separability function.

Twisted conjugacy separability domination conjecture. For every automorphism $\varphi\in\operatorname{Aut}(N)$,

$$\operatorname{Conj}_{N,S}^{\varphi}(n)\preceq\operatorname{Conj}_{N,S}(n).$$

The conjecture asserts that twisted conjugacy is no harder to separate effectively than ordinary conjugacy for nilpotent groups. The paper states that this is open in general.