The generating-ideal conjecture for weak polynomial identities of the Weyl algebra

Let $\mathbb{F}$ be a field of characteristic zero. The Weyl algebra $\mathsf{A}_1$ is generated by $x,y$ with relation $yx=xy+1$, and let $\mathsf{V}=\mathbb{F}\text{-}\operatorname{span}\{x,y\}$. Let $\mathbb{F}\langle X\rangle$ be the free associative algebra, and let $\mathcal{I}$ be the L-ideal generated by

$$\Gamma_3(x_i,x_j,x_k),\qquad {\rm St}_3(x_i,x_j,x_k),\qquad T_4(x_i,x_j,x_k,x_l)$$

for all positive indices. A weak polynomial identity for $(\mathsf{A}_1,\mathsf{V})$ is an element of $\mathbb{F}\langle X\rangle$ vanishing under substitutions from $\mathsf{V}$ into $\mathsf{A}_1$.

Generating-ideal conjecture. The ideal of all weak polynomial identities for the pair $(\mathsf{A}_1,\mathsf{V})$ is equal to $\mathcal{I}$.

The paper establishes the claim for two variables and for degrees four and five, while the conjecture concerns weak polynomial identities in all numbers of variables and degrees.