Two-pass adversarial-order Max-Cut requires linear space

Prove that for every ε > 0, any two-pass adversarial-order streaming algorithm that outputs a (1/2 + ε)-approximation to the Max-Cut value on an undirected graph given as an edge stream must use Ω(n) bits of memory.

Background

The recent breakthrough of FMW25 shows that any k-pass, s-space streaming algorithm achieving (1/2+ε)-approximation for Max-Cut must satisfy ks = Ω(n^{1/3}). It remains unclear how the hardness scales with few passes and larger space. This conjecture strengthens the single-pass Ω(n) lower bound to the two-pass setting, asserting that linear space remains necessary even with one additional pass.