An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod p$ cohomology of GL(n,ℤ)

Let p be a prime number and F an algebraic closure of the finite field Fp with p elements. Let n and N denote positive integers, N prime to p. We are interested in representations of the Galois group GQ = Gal ( ¯ Q/Q) into GL(n, F), unramified at all finite primes not dividing pN. (We shall say