I'm faced with the following problem on primes. Does someone have any clue? Is it (a reformulation of) an open problem?

Let $d$ be a positive integer, $d\geq 2$. By Dirichlet's theorem, there is an infinite set $\mathcal{P}$ of primes congruent to $1$ modulo $d$.

Consider the set $N$ of integers $n=1+kd$ where all prime divisors of $k$ belong to $\mathcal{P}$.

Could we expect that $N$ contains infinitely many primes? (we need at least $d$ to be even).