Hamiltonicity conjecture for randomly perturbed graphs with geometric randomness

Let $dgeq1$ be an integer, let $\alpha=\alpha(n)\in(0,1/2)$, and let $H$ be an $n$-vertex graph with minimum degree at least $\alpha n$. Write $G^d(n,r)$ for the random geometric graph generated by $n$ independent uniform points in $[0,1]^d$, with edges between points at distance at most $r$. Hamiltonicity conjecture. For some constant $C$, asymptotically almost surely,

$$H\cup G^d(n,(-C\log\alpha/n)^{1/d})$$

is Hamiltonian. This would extend the known threshold for binomial random perturbations to random geometric graphs and would be best possible up to the value of $C$; the conjecture concerns the regime in which $\alpha=\alpha(n)$ tends to $0$.