I'm sorry this isn't much of an answer, but you might be interested in this paper of De Koninck, Kátai and Subbarao. Your conjecture is almost strong enough to mean that for any squarefree $a$, there are infinitely many primes $p$ where the squarefree part of $p-d$ is exactly $a$ (the difference being that you do not require $(a,mn)=1$).

Section 4 in the cited paper establishes a sort of converse formulation: for any powerful $m$, there are infinitely many primes $p$ where the powerful part of $p-1$ is exactly $K$ (with the expected asymptotics). Of course this is a far denser set in which to look for primes, so this is certainly a much easier problem. Presumably no significant complications arise if we shift this by $d$ rather than $1$.

I'm sorry this isn't much of an answer, but you might be interested in this paper of De Koninck, Kátai and Subbarao. Your conjecture is almost strong enough to mean that for any squarefree $a$, there are infinitely many primes $p$ where the squarefree part of $p-d$ is exactly $a$ (the difference being that you do not require $(a,mn)=1$).

Section 4 in the cited paper establishes a sort of converse formulation: for any powerful $K$, there are infinitely many primes $p$ where the powerful part of $p-1$ is exactly $K$ (with the expected asymptotics). Of course this is a far denser set in which to look for primes, so this is certainly a much easier problem. Presumably no significant complications arise if we shift this by $d$ rather than $1$.