The $L^{n-1}$-curvature conjecture for two-convex mean curvature flows

Let $\{M_t\subset\mathbb{R}^{n+1}\}_{t\in [0,T)}$ be a mean curvature flow of two-convex closed embedded hypersurfaces. Here $\lvert A\rvert$ denotes the norm of the second fundamental form and $d\mu$ the induced measure on $M_t$. $L^{n-1}$-curvature conjecture. There is a constant $C<\infty$, depending only on the geometric parameters of the initial hypersurface, such that

$$\int_{M_t} \lvert A\rvert^{n-1}\,d\mu\leq C.$$

The conjecture strengthens the known uniform bounds with exponent $n-1-\varepsilon$ for every $\varepsilon>0$ and would provide critical curvature control near the first singular time. Its precise status is not established in the supplied source.