McKean-type blowup criterion for radial EPDiff equations

Let $\omega_0$ be the initial radial momentum, let $r\geq 0$, and consider a $C^1$ radial solution of the Euler–Arnold equation with parameter $k$.

McKean-type blowup conjecture. If

$$\int_r^{\infty}\omega_0(s)\,ds<0$$

for some $r\geq 0$, then a $C^1$ solution with $k=1$, corresponding to the $\dot H^1$ or $H^1$ metric, breaks down in finite time. If instead

$$\int_r^{\infty}s^2\omega_0(s)\,ds<0$$

for some $r\geq 0$, then a $C^1$ solution with $k=2$ breaks down in finite time.

This proposes a higher-dimensional radial analogue of McKean-type sign criteria known for the Hunter–Saxton and Camassa–Holm equations. The source motivates the two weighted integral conditions but does not establish them, so the criterion remains open.