The minimal-intensity conjecture for finite-time Euler singularities

Let $T_*<\infty$ and let $\omega\in C^\infty_c(\mathbb{R}^3\times[0,T_*))$ be a smooth solution of the 3D Euler equation that becomes singular as $t\to T_*$. Minimal-intensity conjecture. There exists a universal finite exponent $p_*$ such that

$$\sup_{t\in[0,T_*)}\|\omega(t)\|_{L^{p_*}}=+\infty.$$

The conjecture would show that every finite-time singularity has a universal minimum intensity in an $L^{p_*}$ norm, potentially without resolving the full blow-up problem.