Tensor factorizations of a unitary operator with simple Lebesgue spectrum

We show that for all $n,p>1$, there exists a unitary operator $U$ such that the tensor product $U\otimes U^p\otimes\dots\otimes U^{p^{n-1}}$ is a unitary operator with simple Lebesgue spectrum. Moreover, there exists an ergodic automorphism $T$ such that the spectrum of $T\odot T$ is simple, while the spectrum of $T\otimes T\otimes T$ is absolutely continuous.