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The selection of an optimal pacing site, which is ideally scar-free and late activated, is critical to the response of cardiac resynchronization therapy (CRT). Despite the success of current approaches formulating the detection of such late mechanical activation (LMA) regions as a problem of activation time regression, their accuracy remains unsatisfactory, particularly in cases where myocardial scar exists. To address this issue, this paper introduces a multi-task deep learning framework that simultaneously estimates LMA amount and classify the scar-free LMA regions based on cine displacement encoding with stimulated echoes (DENSE) magnetic resonance imaging (MRI). With a newly introduced auxiliary LMA region classification sub-network, our proposed model shows more robustness to the complex pattern cause by myocardial scar, significantly eliminates their negative effects in LMA detection, and in turn improves the performance of scar classification. To evaluate the effectiveness of our method, we tests our model on real cardiac MR images and compare the predicted LMA with the state-of-the-art approaches. It shows that our approach achieves substantially increased accuracy. In addition, we employ the gradient-weighted class activation mapping (Grad-CAM) to visualize the feature maps learned by all methods. Experimental results suggest that our proposed model better recognizes the LMA region pattern.