NP-completeness of higher graph powers of girth $2k+1$

Let $k$ and $r$ be positive integers. The problem \textsc{$k^{th}$Power of a Graph With Girth $r$} asks, given a graph $G$, whether there exists a graph $H$ of girth $r$ such that $G=H^k$. Higher-power girth conjecture. The problem \textsc{$k^{th}$Power of a Graph With Girth $r$} for $r=2k+1$ is NP-complete. The conjecture proposes the missing NP-complete case in a complete girth-parametrized complexity dichotomy for higher graph-root finding; related results establish NP-completeness when $r=k$ and a polynomial-time algorithm for finding all $k^{th}$ roots of girth $2k+3$.