Finite-graph conjecture for gradient reaction-diffusion systems

Consider the reaction-diffusion system

$$\begin{cases} \dot u^1=\Delta u^1+f^1(u^i),\\ \vdots\\ \dot u^k=\Delta u^k+f^k(u^i). \end{cases}$$

Assume that the vector field $(f^1,\ldots,f^k)$ is a gradient vector field.

Finite-graph conjecture. The graph of this system has finitely many nodes, and every node is a fixed point, that is, an equilibrium solution.

This predicts that gradient structure rules out non-equilibrium recurrent nodes and permits only finitely many such equilibria. The supplied text gives no evidence that the claim has been proved or refuted.