We study hypothesis testing for penalized estimators in settings where the full marginal distribution of a multivariate response is difficult to specify, such as longitudinal data with correlated measurements or high-dimensional heteroscedastic regression. Assuming that the conditional mean model is correctly specified, we establish that the penalized estimating equations admit a $\sqrt{n}$-consistent solution, even when the working covariance structure is misspecified. Our inferential target is a low-dimensional subvector of parameters associated with the mean model. We show that the resulting test statistic converges to a $χ^2$ distribution, and that its asymptotic power depends on the nuisance covariance function. To mitigate this dependence, we propose estimating the covariance function via cross-fitting, which provides a calibrated and robust procedure for inference.