Ancient-solution triviality conjecture below the Sobolev exponent

Let $(M,g)$ be a complete, $n$-dimensional Riemannian manifold with bounded geometry. An ancient, positive solution is a positive solution $u$ of

$$u_t=\Delta u+u^p$$

that is defined for all past times. Write

$$p_S=\frac{n+2}{n-2}$$

when $n\geq 3$. Ancient-solution triviality conjecture. If $p<p_S$ when $n\geq 3$, or if $p>0$ when $n\leq 2$, then every ancient, positive solution of $u_t=\Delta u+u^p$ is trivial.

This extends the known Euclidean result to complete manifolds with bounded geometry. The statement is presented as a conjecture, and its general validity remains open.