The residue classification conjecture for Mersenne primes

Let $\rho$ be a prime and let $M_\rho=2^\rho-1$. The Mersenne-prime residue conjecture. If $M_\rho$ is a Mersenne prime, then exactly one of the following holds: $M_\rho\rightthreeheadleftarrow 3,7,31,127,247,271\rightthreeheadleftarrow 360$, equivalently, the residue of $M_\rho$ modulo $360$ belongs to \\{3,7,31,127,247,271\}. The claim concerns the possible congruence classes of Mersenne primes modulo $360$; the source gives no resolution evidence.