Is any of the assumptions hold true for the mentioned?
Regularization will always increase the MSE loss on the training set (training loss).  Intuitively, it constraints/limits the set of regression lines you can use.
However, the MSE loss on the test set can be either increase (for the same reason) or decrease (if it has helped combat overfitting and helped bias the classifier towards a simpler hypothesis, and if simpler hypotheses are more likely to be correct).