Higher Sobolev boundedness conjecture for global cubic-wave solutions

Let $u$ solve the cubic wave equation on $[0,\infty)$ with initial data $(u_0,u_1)\in\mathcal{H}^{1/2}\cap\mathcal{H}^1$, where $\boldsymbol{u}(t)=(u(t),\partial_tu(t))$ and $\mathcal{H}^s=\dot H^s(\mathbb{R}^3)\times\dot H^{s-1}(\mathbb{R}^3)$. Higher Sobolev boundedness conjecture. For every $\nu\in(1/2,1]$,

$$\limsup_{t\to\infty}\|\boldsymbol{u}(t)\|_{\mathcal{H}^{\nu}}<\infty.$$

The conjecture concerns the regularity of solutions with asymptotic self-similar behaviour and asserts boundedness of every Sobolev norm strictly above the critical index. The source gives no resolution.