Borrowing external controls to augment the concurrent control arm is a popular topic in clinical trials. Bayesian dynamic borrowing methods adaptively discount external controls according to prior-data conflict. For the Gaussian endpoint, parameter-specific information borrowing enables differential discounting between the population mean and variance. The borrowing-by-part power prior employs two power parameters to separately downweight external likelihoods concerning the sample mean and variance. However, within the fully Bayesian framework, the posterior inference of the average treatment effect (ATE) defined as the population mean difference is significantly affected by the variance-specific prior-data conflict that reflects the heterogeneity of population variance. Here, we propose the modified conditional borrowing-by-part power prior (MCBPP) that separately discounts the external sample mean and variance according to parameter-specific prior-data conflicts, resulting in a more stable posterior estimation of ATE than its competitors under the same degree of mean-specific prior-data conflict. By fully discounting the external sample variance, the robust MCBPP (rMCBPP) can yield robust posterior inference of ATE against the variance-specific prior-data conflict. Although the population variance is considered a nuisance parameter, its homogeneity is equally important to justify information borrowing. We recommend the rMCBPP for borrowing external controls with a similar sample variance to concurrent controls because it exhibits better control of bias and Type I error rate than the modified power prior (MPP) assuming unknown variance in the absence of population variance heterogeneity. However, when faced with a significant sample variance discrepancy, the MPP assuming unknown variance is preferred given its better performance under severe population variance heterogeneity.