Upper-bound conjecture for the Gaussian free field reconstruction threshold

Let $T_{\mathrm{rec}}^+$ denote the upper reconstruction threshold for the statistical reconstruction problem considered in the paper. Upper-threshold conjecture.

$$T_{\mathrm{rec}}^+\leq 2\sqrt{2\pi}.$$

The conjecture is motivated by the analysis of Gaussian-free-field level lines and is presented as an upper bound on the reconstruction threshold. No proof or resolution is supplied in the surrounding text.