Finite soluble groups satisfying the swap conjecture

For a d-generated finite group G, we consider the graph Δd(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _d(G)$$\end{document} (swap graph) in which the vertices are the ordered generating d-tuples and in which two vertices (x1,…,xd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_1,\ldots ,x_d)$$\end{document} and (y1,…,yd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y_1,\ldots ,y_d)$$\end{document} are adjacent if and only if they differ only by one entry. It was conjectured by Tennant and Turner that Δd(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _d(G)$$\end{document} is a connected graph. We prove that this conjecture is true if G is a soluble group satisfying some extra conditions, for example if the derived subgroup of G has odd order or is nilpotent.