Heteroclinic-orbit conjecture for eternal solutions

Let $u$ be an eternal solution of the semilinear parabolic equation $$, with coefficients $a_i$ that decay sufficiently fast as $|x|\to\infty$. Heteroclinic-orbit conjecture. All eternal solutions are heteroclinic orbits. This is presented as a more ambitious strengthening of the universal-bound conjecture, requiring control of the $1$-norm as well; its status is not resolved in the source.