Sharpness conjecture for the singular-time dimension of wild Euler solutions

Let $T>0$, let $\beta<\frac{1}{3}$, and let $v$ be a non-conservative weak solution of the incompressible Euler equations on $\mathbb{T}^3\times[0,T]$. For a closed set $\mathcal{B}\subset[0,T]$, assume that $v$ is smooth on $\mathbb{T}^3\times\mathcal{B}^c$.

Sharpness conjecture. For every $\beta<\frac{1}{3}$, there exists a non-conservative weak solution

$$v\in C^\beta\left(\mathbb{T}^3\times[0,T]\right)$$

and a closed set $\mathcal{B}\subset[0,T]$ such that

$$v\in C^\infty\left(\mathbb{T}^3\times\mathcal{B}^c\right),\qquad \mathop{\rm \dim}\nolimits_{\mathcal{H}}(\mathcal{B})=\frac{2\beta}{1-\beta}.$$

The preceding lower bound shows that this Hausdorff dimension would be optimal. Establishing existence at the critical dimension would prove sharpness of the size estimate for singular times of non-conservative Hölder Euler solutions.