Espuny Díaz and Girão's Hamiltonicity threshold conjecture for random clique-factors

For a fixed integer $r\ge 2$, let $K_r$ denote the complete graph on $r$ vertices, and let $\alpha^*(K_r)$ be the minimum-degree threshold of Hamiltonicity for perturbation by a uniformly random $K_r$-factor. Espuny Díaz and Girão's conjecture. For all $r\ge 2$, $\alpha^*(K_r)$ is the unique real positive solution to

$$x^r+rx-1=0.$$

This conjecture generalizes the known case $\alpha^*(K_2)=\sqrt{2}-1$, and predicts the Hamiltonicity threshold for random clique-factor perturbations for every fixed clique size. Its status is not resolved in the supplied source.