The twisted conjugacy problem for pairs of endomorphisms in nilpotent groups

An algorithm is constructed that, when given an explicit presentation of a finitely generated nilpotent group $G,$ decides for any pair of endomorphisms $\varphi, \psi : G \to G$ and any pair of elements $u, v \in G,$ whether or not the equation $(x\varphi)u = v (x\psi)$ has a solution $x \in G.$ Thus it is shown that the problem of the title is decidable. Also we present an algorithm that produces a finite set of generators of the subgroup (equalizer) $Eq_{\varphi, \psi}(G) \leq G$ of all elements $u \in G$ such that $u \varphi = u \psi .$