Global unconditional well-posedness conjecture for the defocusing CCM equation

Let $L^2_{+}(\mathbb{T})$ and $L^2_{+}(\mathbb{R})$ denote the positive-frequency Hardy subspaces of $L^2(\mathbb{T})$ and $L^2(\mathbb{R})$, respectively. Consider the defocusing CCM equation on each of these domains. Global unconditional well-posedness conjecture. The defocusing CCM equation is unconditionally globally well-posed in

$$L^2_{+}(\mathbb{T})$$

and

$$L^2_{+}(\mathbb{R}).$$

This conjecture concerns unconditional uniqueness at the scaling-critical regularity, in addition to global existence and continuous dependence in the indicated Hardy spaces. The preceding discussion presents the estimate supporting the conjecture, but no resolution is given here.