Multidimensional configurations in the primes with shifted prime steps

Let $\mathcal{P}$ denote the set of primes. For a fixed dimension $d$, Cook-Magyar-Titichetrakun, Tao-Ziegler and Fox-Zhao independently proved that any subset of positive relative density of $\mathcal{P}^d$ contains an arbitrary linear configuration. In this paper, we prove that there exists such configuration with the step being a shifted prime (prime minus $1$ or plus $1$).