For Gaussian random fields observed on a fixed domain, the regression slope is consistently estimable when the exposure is not too smooth relative to the confounder, and a spectral ratio condition is necessary.
Asymptotically equivalent prediction in multivariate geostatistics.Bernoulli, 28(4): 2518–2545, 2022
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Consistent Infill Estimability of the Regression Slope Between Gaussian Random Fields Under Spatial Confounding
For Gaussian random fields observed on a fixed domain, the regression slope is consistently estimable when the exposure is not too smooth relative to the confounder, and a spectral ratio condition is necessary.