Where competition stops compressing the spread

Two market makers compete for the same flow. Each faces the same costs: adverse selection plus fees add up to a $4$-cent cost floor per fill (i.e., a fill loses money unless the half-spread is at least $4$ cents). The tick is \0.01$. They repeatedly undercut each other to win the queue.

Where does the undercutting stop? What spread do we end up with, and why isn't it zero?