  HD$\,$169142 is an excellent target to investigate signs of planet-disk
interaction due to the previous evidence of gap structures. We performed J-band
(~1.2{\mu}m) polarized intensity imaging of HD169142 with VLT/SPHERE. We
observe polarized scattered light down to 0.16" (~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" (~21 au) and 0.56" (~66 au),
and marginally detect a faint third gap at 0.70"-0.73" (~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. It, however, 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 pile-up
of millimeter grains in a dust trap and the continuous distribution of small
grains throughout the gap likely require a 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.
