The classification conjecture for non-nilpotent evolution algebras satisfying condition P

Let $E$ be an $n$-dimensional non-nilpotent evolution algebra satisfying condition $P$. Let $ES_1$ denote the one-dimensional evolution algebra used in the source, let $\mathbb{C}^{m}$ denote the $m$-dimensional zero algebra, and let $\widetilde{E}\in ZN^{s}$ be a nilpotent evolution algebra with maximal index of nilpotency. Classification conjecture. The algebra $E$ is isomorphic to one of the following pairwise non-isomorphic algebras:

$$ES_1\oplus\mathbb{C}^{n-1},\qquad ES_1\oplus\widetilde{E}\oplus\mathbb{C}^{n-s-1}.$$

The claim is presented as the second conjecture and as a consequence of the preceding nonvanishing-solution conjecture. If the first conjecture holds, the paper states that this description follows.