Kernel ideal conjecture for the two-differential form map

Let $A_n$ be the free associative algebra and let $\varphi(d,D):A_n\to\Omega(d,D)^{even}_*$ be the algebra map sending each generator $x_i$ to $x_i$. For $m_1,m_2,m_3,m_4\in A_n$, define

$$R(m_1,m_2,m_3,m_4)=[m_4,[m_3,[m_1,m_2]]]-[m_1,[m_3,[m_4,m_2]]]+[m_2,[m_3,[m_1,m_4]]].$$

The source uses the six-term relation $R=R(m_1,m_2,m_3,m_4)-R(m_1,m_3,m_2,m_4)$. Kernel ideal conjecture. The kernel of $\varphi(d,D)$ is the two-sided ideal $A_n\cdot\{R\}\cdot A_n$.