HD 169142 is an excellent target for investigating signs of planet-disk interaction due to previous evidence of gap structures. We perform J-band (similar to 1.2 mu m) polarized intensity imaging of HD 169142 with VLT/SPHERE. We observe polarized scattered light down to 0 ''.16 (similar to 19 au) and find an inner gap with a significantly reduced scattered-light flux. We confirm the previously detected double-ring structure peaking at 0 ''.18 (similar to 21 au) and 0 ''.56 (similar to 66 au) and marginally detect a faint third gap at 0 ''.70-0 ''.73 (similar to 82-85 au). We explore dust evolution models in a disk perturbed by two giant planets, as well as models with a parameterized dust size distribution. The dust evolution model is able to reproduce the ring locations and gap widths in polarized intensity but fails to reproduce their depths. However, it gives a good match with the ALMA dust continuum image at 1.3 mm. Models with a parameterized dust size distribution better reproduce the gap depth in scattered light, suggesting that dust filtration at the outer edges of the gaps is less effective. The pileup of millimeter grains in a dust trap and the continuous distribution of small grains throughout the gap likely require more efficient dust fragmentation and dust diffusion in the dust trap. Alternatively, turbulence or charging effects might lead to a reservoir of small grains at the surface layer that is not affected by the dust growth and fragmentation cycle dominating the dense disk midplane. The exploration of models shows that extracting planet properties such as mass from observed gap profiles is highly degenerate.