In the answer to this question, engelbret mentions "a conjecture of H. L. Montgomery (not the one on pair correlations, another one), which implies both the GRH and the ElliottHalberstam conjecture.". My question is : what does this conjecture says, and where is it stated ? I am also interested in any references that would discuss that conjecture.

I think the conjecture you have in mind is that for an arithmetic progression $a\pmod q$ with $(a,q)=1$ and $x\ge q$ $$ \Psi(x;q,a) = \frac{x}{\phi(q)} + O\Big(\frac{x^{\frac 12+\epsilon}}{\sqrt{q}}\Big). $$ This is stronger than GRH which has the error $O(x^{\frac 12+\epsilon})$, and just summing the errors over all $q\le x^{1\epsilon}$ you get the ElliottHalberstam conjecture as well. 


discussion: Limitations to the equidistribution of primes (Friedlander and Granville, 1989) and Small gaps between prime numbers (Soundararajan, 2007) (page 16). the sources to Montgomery are Topics in multiplicative number theory (Springer, 1971) Proc. Symp. Pure Math. 28, 307 (1976) 

