The solvability–idempotent conjecture for complex evolution algebras

Let $\mathcal{E}$ be a complex evolution algebra. An element $u\in\mathcal{E}$ is an idempotent if $u^2=u$. If $\mathcal{E}$ has natural basis $\{e_1,\dots,e_n\}$ and structure matrix $M_B(\mathcal{E})$, then idempotents correspond to solutions of

$$M_B(\mathcal{E})^t\begin{pmatrix}x_1^2\\ \vdots\\ x_n^2\end{pmatrix}=\begin{pmatrix}x_1\\ \vdots\\ x_n\end{pmatrix}.$$

The algebra is solvable if $\mathcal{E}^{(k)}=0$ for some $k$, where $\mathcal{E}^{(1)}=\mathcal{E}$ and $\mathcal{E}^{(k+1)}=\mathcal{E}^{(k)}\mathcal{E}^{(k)}$. Solvability–idempotent conjecture. The following assertions are equivalent: $\mathcal{E}$ is solvable; $\mathcal{E}$ admits no idempotents; and the displayed system only admits the trivial solution. The absence of idempotents is immediately necessary for solvability, while existence of idempotents in arbitrary evolution algebras is stated to remain open. The conjecture proposes that this necessary condition is also sufficient, equivalently that nonsolvable algebras possess a nontrivial idempotent.