Given only that E ≥ F = G ≥ H, which of the following is definitely true?

Compose symbol by symbol: E ≥ F, F = G, G ≥ H. An equality passes a relation straight through, so the chain is really E ≥ G ≥ H, and a non-strict link followed by a non-strict link composes to a non-strict result: E ≥ H. Nothing stronger is available, because a strict conclusion needs at least one strict link somewhere in the chain and there is none here — E = F = G = H satisfies every statement. That single example is what kills 'E > H', the trap for a solver who reads a descending run of ≥ signs as though it produced >. 'E = H' fails on E = 9, F = G = H = 2, and 'H > E' reverses the chain outright.