The below-scaling ill-posedness conjecture for semilinear wave equations

For a semilinear wave equation, let $H^s \times H^{s-1}$ denote the Sobolev data space, and let $s_0$ be the scaling index determined by the equation's scaling. Below-scaling ill-posedness conjecture. Semilinear wave equations are ill-posed below scaling, that is, for $s<s_0$. Scaling heuristics suggest that a small-data, small-time result below the scaling index would rescale into a large-data, large-time result, but no proof of this ill-posedness was known to the author.