The infinite-dimensionality conjecture for integrable evolution equations

Let $\mathcal{E}$ be the infinite prolongation of a $\textup{(1+1)}$-dimensional evolution equation. For $p\in\mathbb{Z}_{\ge 0}$ and $a\in\mathcal{E}$, let $\operatorname{\mathbb{F}}^{p}(\mathcal{E},a)$ denote the corresponding Lie algebra. Infinite-dimensionality conjecture. If the equation is integrable, then there exist $p\in\mathbb{Z}_{\ge 0}$ and $a\in\mathcal{E}$ such that $\operatorname{\mathbb{F}}^{p}(\mathcal{E},a)$ is infinite-dimensional and, for every nilpotent ideal $\mathfrak{I}\subset\operatorname{\mathbb{F}}^{p}(\mathcal{E},a)$, the quotient

$$\operatorname{\mathbb{F}}^{p}(\mathcal{E},a)/\mathfrak{I}$$

is infinite-dimensional as well. The conjecture proposes an algebraic criterion for detecting integrability: an integrable equation must have a prolongation Lie algebra with an infinite-dimensional non-nilpotent quotient.