The most fundamental undecidable question on tilings is the Domino Problem that asks whether a Wang tileset tiles the discrete plane. Lukkarila proved in 2009 that it remains undecidable when restricting the input to the class of 4-way deterministic tilesets. Due to the existence of aperiodic tilesets, the most natural distinct variant of this problem is the Periodic Domino Problem which asks whether a Wang tileset admits a periodic tiling of the plane. This problem is also undecidable. Jeandel recently discovered a new and elegant proof for this result. Inspired by this new proof technique and some ingredients from Lukkarila's construction, we prove that it remains undecidable when restricted to 4-way deterministic tilesets.