Conjecture on the unique blowup point for Fourier-restricted Euler and hypodissipative Navier–Stokes

Let $u\in C\left([0,T_{\max}); \dot{H}^s_{\mathcal{M}}\right)$, with $s>\frac{5}{2}$, be an odd, permutation symmetric, $\sigma$-mirror symmetric solution of the Fourier-restricted Euler or hypodissipative Navier–Stokes equation with finite-time blowup as in the stated blowup results. Unique blowup-point conjecture. The origin is the only blowup point, and for every $\epsilon>0$,

$$\nabla u\in L^\infty\left([0,T_{\max})\times\{x\in\mathbb{T}^3:|x|>\epsilon\}\right).$$

The preceding discussion explains that the cosine-series representation of $\nabla u$ gives precise information at the origin, while substantial cancellation is expected away from the origin. Establishing uniqueness of the blowup point would require delicate harmonic analysis and is not proved in the source.