Lower-bound conjecture for the Gaussian free field reconstruction threshold

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

$$T_{\mathrm{rec}}^-\geq 2\sqrt{\pi}.$$

The claim is presented as the objective of the surrounding section, which develops intermediate results supporting a lower bound on the reconstruction regime. No resolution is given in the supplied text.