Polynomial configurations in the primes

The Bergelson-Leibman theorem states that if P_1, …, P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), …, x+P_k(m), where x,m are integers. Various generalizations of this theorem are known. Wooley and Ziegler showed that the variable m can in fact be taken to be a prime minus 1, and Tao and Ziegler showed that the Bergelson-Leibman theorem holds for subsets of the primes of positive relative upper density. Here we prove a hybrid of the latter two results, namely that the step m in the Tao-Ziegler theorem can be restricted to the set of primes minus 1.