The presentation conjecture for the even subalgebra of the TAMA

Let $U(\mathfrak{so}(n))$ be the universal enveloping algebra, and let $\mathfrak{O}_{\overline{0}}$ denote the even subalgebra of the TAMA. For a sequence $A=(a_1,a_2,a_3,a_4)$ of four distinct indices, write

$$X_A=X_{a_1a_2}X_{a_3a_4}+X_{a_1a_3}X_{a_4a_2}+X_{a_1a_4}X_{a_2a_3},$$

where $X_{ij}=E_{ij}-E_{ji}\in U(\mathfrak{so}(n))$. Let $I$ be the ideal in the exact sequence

$$0\to I\to U(\mathfrak{so}(n))\to\mathfrak{O}_{\overline{0}}\to0.$$

For sequences $A=(a_1,a_2,a_3,a_4)$ and $B=(b_1,b_2,b_3,b_4)$ of four distinct indices, and with $\mathcal{T}$ defined as in the source, the presentation conjecture for the even subalgebra of the TAMA. The ideal $I$ is generated by the elements

$$\sum_{\sigma\in\mathcal{T}}\sigma\left(X_AX_B+\sum_{(a,b)\subset A\cap B}X_{A\setminus(a,b)}X_{B\setminus(a,b)}\right)-12\delta_{A,B},$$

and consequently

$$\mathfrak{O}_{\overline{0}}\cong U(\mathfrak{so}(n))\bigg/\left\langle\sum_{\sigma\in\mathcal{T}}\sigma\left(X_AX_B+\sum_{(a,b)\subset A\cap B}X_{A\setminus(a,b)}X_{B\setminus(a,b)}\right)-12\delta_{A,B}\right\rangle.$$

This conjecture proposes an explicit presentation of the even TAMA in terms of generators and relations; the supplied text gives no evidence that the presentation has been proved or disproved.