The prediction of petrophysical properties in multi-phase rock is critical for various geoenergy applications. To this end, deep learning-based methods have recently emerged as a prominent research focus in rock physics. However, due to the inherent scarcity of structural data in multi-phase rocks, coupled with the limitations of convolutional neural network in capturing multi-scale dynamic phase interface and mitigating parameter interference in multi-task learning, predictive performance has not yet reached a satisfactory level. To address this shortcoming, a new multi-task learning framework based on a multi-gate mixture-of-different-experts model is proposed to predict multiple properties of multi-phase rocks. A semi-supervised fractal-informed generative adversarial network is employed to reconstruct multi-phase rocks images, while the finite element method is used to compute their transport properties. The gating network allocates bespoke expert subsets to each task, and an automatic weighted loss function dynamically balances the task-specific loss contributions, enhancing performance and generalization. The results show that the statistical average of the predicted relative permeability aligns with the Brooks-Corey equation, demonstrating the reliability of the proposed model in preserving fundamental physical principles. On the validation dataset, the model achieves high predictive accuracy across all target properties, including fractal dimension, porosity, saturation, permeability of gas phase, permeability of water phase, and effective permeability. Comparative evaluation on the test dataset demonstrates that the proposed model significantly outperforms other multi-task models. These findings confirm that the proposed framework can simultaneously and accurately predict multi-phase rock properties under limited data conditions, holding a promise for guiding assessments in hydrocarbon and geothermal exploration, CO₂ sequestration, nuclear waste disposal, and geological hazard mitigation, among others.