Wolf's semisimplicity conjecture for deformed Fomin–Kirillov algebras

Let $K$ be an algebraically closed field of characteristic zero, let $\alpha_1,\alpha_2\in K$, and let $\mathcal{D}_4(\alpha_1,\alpha_2)$ be the quadratic $K$-algebra generated by $x_{ij}=-x_{ji}$ for $1\leq i\neq j\leq 4$, with relations

$$x_{ij}^2=\alpha_1,$$ $$x_{ij}x_{kl}-x_{kl}x_{ij}=0,$$ $$x_{ij}x_{jk}+x_{jk}x_{ki}+x_{ki}x_{ij}=\alpha_2$$

for the corresponding distinct indices. Wolf's conjecture. The algebra $\mathcal{D}_4(\alpha_1,\alpha_2)$ is semisimple if and only if

$$(\alpha_1-\alpha_2)(\alpha_1+\alpha_2)\neq 0.$$

Wolf proved semisimplicity under the stronger condition $\alpha_1(3\alpha_1-\alpha_2)(\alpha_1-\alpha_2)(\alpha_1+\alpha_2)\neq 0$; the asserted converse and the remaining parameter cases are therefore the subject of the conjecture.