statistics considered in @BM2016, this type of testing problem can be extended to the sparsity test for the residual processes of a continuous-time factor model with an observable factor thus promising in applications to high-frequency financial data. In addition, such a problem will also be useful for covariance matrix modeling in a low-frequency setting because it often suffers from the curse of dimensionality due to the increase of the number of unknown parameters to be estimated, and thus it is a common practice to impose a certain structure on covariance matrices for reducing the number of unknown parameters in models. For example, @TWYZ2011 have proposed fitting a matrix factor model to daily covariance matrices which are estimated from high-frequency data using the methodology of [@WZ2010], while @KO2016 [@KO2018] have introduced a dynamic (multiple-block) equicorrelation structure to multivariate stochastic volatility models. The afore-mentioned testing will be useful for examining the validity of such specification. If the dimension $d$ is fixed, the desired approximation can be obtained as a simple consequence of a multivariate mixed-normal limit theorem for $\sqrt{n}(\widehat{[Y,Y]}^n_1-[Y,Y]_1)$, which is well-studied in the literature and holds true under quite mild assumptions; see e.g. Theorem 5.4.2 of [@JP2012]. The problem here is 