Grothendieck's period conjecture for Kummer surfaces of self-product CM type

We show that the Grothendieck period conjecture holds for the Kummer surface associated with the square of a CM elliptic curve. This means that the period isomorphism is dense in the torsor of motivic periods. In other words, the isomorphism is dense in the torsor of motivated periods, and motivated classes on powers of the surface are algebraic. The point is that the motive has a non-trivial transcendental part, but belongs to the Tannakian category generated by the motive of a CM elliptic curve.