Massive key performance indicators (KPIs) are monitored as multivariate time series data (MTS) to ensure the reliability of the software applications and service system. Accurately detecting the abnormality of MTS is very critical for subsequent fault elimination. The scarcity of anomalies and manual labeling has led to the development of various self-supervised MTS anomaly detection (AD) methods, which optimize an overall objective/loss encompassing all metrics’ regression objectives/losses. However, our empirical study uncovers the prevalence of conflicts among metrics’ regression objectives, causing MTS models to grapple with different losses. This critical aspect significantly impacts detection performance but has been overlooked in existing approaches. To address this problem, by mimicking the design of multi-gate mixture-of-experts (MMoE), we introduce CAD, a Conflict-aware multivariate KPI Anomaly Detection algorithm. CAD offers an exclusive structure for each metric to mitigate potential conflicts while fostering inter-metric promotions. Upon thorough investigation, we find that the poor performance of vanilla MMoE mainly comes from the input-output misalignment settings of MTS formulation and convergence issues arising from expansive tasks. To address these challenges, we propose a straightforward yet effective task-oriented metric selection and p&s (personalized and shared) gating mechanism, which establishes CAD as the first practicable multi-task learning (MTL) based MTS AD model. Evaluations on multiple public datasets reveal that CAD obtains an average F1-score of 0.943 across three public datasets, notably outperforming state-of-the-art methods. Our code is accessible at https://github.com/dawnvince/MTS_CAD.