Local well-posedness conjecture for cubic NLS below $L^2$

Consider the cubic nonlinear Schrödinger equation referred to as, with initial data in the Sobolev space $H^s$. Local well-posedness conjecture. The cubic NLS equation is locally well-posed for initial data in $H^s$ with

$$s\geq -\frac16.$$

The threshold $s=-\frac16$ is motivated by the connection with the modified Korteweg–de Vries equation and by high–low frequency interactions in the analysis. The supplied text does not state whether this conjecture has been resolved.