The $R_\infty$-property for right-angled Artin groups

Given a group $G$ and an automorphism $\varphi$ of $G$, two elements $x, y \in G$ are said to be $\varphi$-conjugate if $x = g y \varphi(g)^{-1}$ for some $g \in G$. The number of equivalence classes is the Reidemeister number $R(\varphi)$ of $\varphi$, and if $R(\varphi) = \infty$ for all automorphisms of $G$, then $G$ is said to have the $R_\infty$-property. A finite simple graph $\Gamma$ gives rise to the right-angled Artin group $A_{\Gamma}$, which has as generators the vertices of $\Gamma$ and as relations $vw = wv$ if and only if $v$ and $w$ are joined by an edge in $\Gamma$. We conjecture that all non-abelian right-angled Artin groups have the $R_\infty$-property and prove this conjecture for several subclasses of right-angled Artin groups.