Finite endomorphic presentation conjecture for Fibonacci Lie algebras

Let $p=2$, let $\mathcal L=\operatorname{Lie}(v_1,v_2)$ be the Fibonacci Lie algebra, and let $\mathbf L=\operatorname{Lie}_p(v_1,v_2)$ be its Fibonacci restricted version. The source proposes that both algebras satisfy the listed defining relations and all their shifts under the shift endomorphism. Finite endomorphic-presentation conjecture. The relations given in Lemma Lrelations define $\mathcal L$ and $\mathbf L$. This is motivated by finite endomorphic presentations of self-similar groups; the displayed relations are verified as relations, but completeness as a defining presentation is left conjectural.