A simple dialogical characterization of the logic LQ of weak excluded middle?

I've recently returned to a problem about dialogues and the propositional logic LQ of weak excluded middle: is it true that if, in Felscher's ruleset E for intuitionistic logic, one changes the rule

Proponent may assert an atom p only if Opponent has already asserted p

to

Proponent may assert an atom p only if Opponent has already either asserted p or asserted ¬p

but leaves all other rules unchanged? If one relaxes the restriction on Proponent's use of atoms as described, then the law of weak excluded middle ¬q∨¬¬q becomes valid, while the usual law of the excluded middle q∨¬q remains invalid. This naturally suggests, in light of the soundness and completeness of the ruleset E for intuitionistic logic, that the modified ruleset characterizes the logic LQ of weak excluded middle, a fascinating superintuitionistic logic. My interest in LQ comes from constructive mathematics, but in any case, the problem is now purely a dialogical one. I'm not invested one way or the other in this result, but I would like to try to show that this simple change in one of Felscher's rules gives rise to an interesting known logic. Of course, a simple counterexample would be nice, too, but so far I've not found any. The problem here is unlike the case of N, which (for the moment) has no known characterization or meaning outside the context of dialogues.