Non-vacuity of improved covariate-adjusted variance

Let $\nu^{\mathsf{rd}}$, $\nu^{\mathsf{rd}\dag}$, and $\mathsf{var}(\widehat{\tau}_{\mathsf{unadj}})$ denote the variance quantities defined in the paper for the covariate-adjusted and unadjusted estimators. The data-generating process for which

$$\nu^{\mathsf{rd}} < \nu^{\mathsf{rd}\dag}$$

and

$$\nu^{\mathsf{rd}} < \mathsf{var}(\widehat{\tau}_{\mathsf{unadj}})$$

is not vacuous.

This asserts that there are data-generating processes under which the relevant covariate-adjusted procedure has smaller variance than both the comparison variance $\nu^{\mathsf{rd}\dag}$ and the variance of the unadjusted estimator. The claim supports the paper's motivation for higher-order covariate adjustment, although the supplied text does not establish whether such a data-generating process has been exhibited or whether the claim remains unresolved.