Infinite-dimensionality conjecture for generalized Fomin–Kirillov algebras

Let $n$ be a positive integer, let $\alpha_{ijk},\beta_{ijk},\gamma_{ij},\lambda_{ijkl}\in\{1,-1\}$, and let $A(\alpha,\beta,\gamma,\lambda)$ be the algebra generated by $x_{ij}$ for $i\ne j$ and $1\leq i,j\leq n$, subject to

$$x_{ij}^2=0,\qquad x_{ij}=\gamma_{ij}x_{ji},$$ $$x_{ij}x_{jk}+\alpha_{ijk}x_{jk}x_{ki}+\beta_{ijk}x_{ki}x_{ij}=0,$$

for $1\leq i,j,k\leq n$, and

$$x_{ij}x_{kl}=\lambda_{ijkl}x_{kl}x_{ij}$$

for any distinct $i,j,k,l$. Infinite-dimensionality conjecture. The algebra $A(\alpha,\beta,\gamma,\lambda)$ is infinite dimensional when $n>4$. Furthermore, the Nichols algebra $\mathfrak{B}(\mathcal O_{(1\ 2)},\rho)$ is infinite dimensional for $n>6$ when $\rho=\operatorname{sgn}\otimes\operatorname{sgn}$ or $\rho=\epsilon\otimes\operatorname{sgn}$. The claim concerns the generalized quadratic algebras extending the Fomin–Kirillov algebra and specific Nichols algebras over the symmetric group; the source gives no resolution or supporting evidence beyond this assertion.