Finite-time blowup conjecture for incompressible Euler flow on compact manifolds

Let $(M,g)$ be a compact Riemannian manifold of dimension $d>2$. Let $u:[0,T_*)\to\Gamma(TM)$ and $p:[0,T_*)\to C^\infty(M)$ be a smooth solution of the incompressible Euler equations

$$\partial_t u^\mathrm{k}+u^\mathrm{j}\nabla_\mathrm{j}u^\mathrm{k}=-\nabla^\mathrm{k}p, \qquad \nabla_\mathrm{k}u^\mathrm{k}=0.$$

Finite-time blowup conjecture. There exists such a compact Riemannian manifold $(M,g)$, of some dimension $d>2$, and such a smooth solution that cannot be smoothly continued to the finite blowup time $T_*<\infty$.

Finite-time singularity formation is a famous open problem even for the three-dimensional flat torus; this conjecture asks for an example on some compact Riemannian manifold and in some dimension greater than two.