Local resilience conjecture for Hamiltonicity in random geometric graphs

Let $G_d(n,r)$ be the $d$-dimensional random geometric graph on $n$ random points, with edges joining pairs at distance at most $r$. A graph is an $(1/2+\epsilon)$-subgraph of $G_d(n,r)$ if it is obtained by retaining, at every vertex, at least a $(1/2+b5)$-fraction of its incident edges.

Local resilience conjecture. For every $\epsilon\in (0,1/2]$ and integer $d\geq1$, there exists a constant $C=C(d,\epsilon)>0$ such that, for $r\geq C(\log n/n)^{1/d}$, asymptotically almost surely every $(1/2+\epsilon)$-subgraph of $G_d(n,r)$ is Hamiltonian.

This would extend the corresponding local resilience results for binomial random graphs and random regular graphs to random geometric graphs. The conjecture is posed as an analogue of known sharp Hamiltonicity results, and its status is open.