$R_{\mathrm{\infty }}$-property for finitely generated torsion-free 2-step nilpotent groups of small Hirsch length

In this paper we will show that finitely generated torsion-free 2-step nilpotent groups of Hirsch length at most 6 do not have the $R_{\mathrm{\infty }}$-property, while there are examples of such groups of Hirsch length 7 that do have the $R_{\mathrm{\infty }}$-property.