Fractional distortion in hyperbolic groups

For all integers \(p>q>0\) and \(k >0\), and all non-elementary torsion-free hyperbolic groups \(H\), we construct a hyperbolic group \(G\) in which \(H\) is a subgroup, such that the distortion function of \(H\) in \(G\) grows like \(\exp^k(n^{p/q})\). Here, \(\exp^k\) denotes the \(k\)-fold-iterated exponential function.