Non-recovery conjecture for geometric planted matchings

Let $n$ be the number of points, $d$ the dimension, $\sigma^2$ the noise parameter, and $\mathcal{E}$ the set of errors made by the maximum-likelihood estimator. Write $d=\omega(\log n)$ when $d/\log n\to\infty$, and use $d\sim a\log n$ for a fixed positive constant $a$. "Non-recovery conjecture." Suppose that any of the following conditions holds:

1. $1\ll d\ll\log n$ and, for some $\epsilon>0$, $\sigma^2\geq n^{-(2-\epsilon)/d}$.
2. $d\sim a\log n$ and, for some $\epsilon>0$, $\sigma^2\geq \frac{1}{(2e^{1/a}-1)^2-1}+\epsilon$.
3. $d=\omega(\log n)$ and, for some $\epsilon>0$, $\sigma^2\geq (\frac14+\epsilon)\frac{d}{\log n}$.

Then, for some $c=c(\epsilon)>0$, $|\mathcal{E}|\geq cn$ with high probability. This conjecture would complete the high-level picture by predicting a macroscopic number of errors in the parameter regimes not covered by the paper's recovery results; the stated thresholds are suggested by first-moment combinatorics of augmenting cycles.