Optimal half-spread

A market maker quotes a bid at fair value minus $x$ and an ask at fair value plus $x$, with $x\ge 0$. In each round the ask is lifted with probability $0.9-0.12x$ and, independently, the bid is hit with the same probability, each trading one unit at the quoted price. Fair value does not change. What is the largest achievable expected profit per round relative to fair value, in the same units as $x$?

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