Unconditional local well-posedness threshold for the periodic generalized KdV equation

Let $k\geq 3$ and consider the $k$-generalized Korteweg-de Vries equation on the periodic domain, with initial data in the Sobolev space $H^s(\mathbb T)$. Unconditional well-posedness conjecture. The equation is unconditionally locally well-posed in $H^s(\mathbb T)$ for every

$$s\geq \frac{1}{2}-\frac{1}{k}.$$

The proposed threshold is motivated by the failure of the convex integration scheme below the regime where the nonlinearity $u^k$ belongs to $C_t^0L_x^1$, and by the known sharp threshold for the KdV equation. The corresponding unconditional nonuniqueness-versus-uniqueness transition remains conjectural for $k\geq 3$.