HINT: We first notice that if for every $(M,d)$ relatively prime there is at least one prime of the form $Mn+d$ then there are infinitely many primes of this form. The first number that every prime $p_i$ sieves is $p_i^2$ so $n_0 +1$ must be greater than $p_i^2/M$( where $n_0$ is the first number of the previous form that $p_i$ could sieve). If a prime divides $Mn+d$ for some number n it divides exactly $M(n+p_i)+d , M(n+2p_i)+d,etc.$. We can see that we have an arithmetic progression of the form $A_i=k_i+np_i$ where $k_i=n_0+1$ . Of course we can give more conditions for the $k_i's$ but the meaning of the question is :Can we prove Dirichlet's theorem using only this condition? This is the reason why my conjecture is a generalisation of Dirichlet's theorem.

Same reasoning gives the modification that I give in my question for the polignac's conjecture, etc.