The determinantal presentation conjecture for the invariant ring

Let $Q$ be the quiver and $I$ the relations defining the representation space in the preceding construction, and let $\EuScript{R}^G=\operatorname{Rep}(\mathbb{C}Q/I,\updelta)/\! /\mathrm{GL}$ be its invariant ring. Let $\varphi:\mathbb{C}[\mathsf{w},\mathsf{v}]\to\EuScript{R}^G$ be the natural homomorphism defined by

$$\mathsf{w}_1\mapsto D_1U_2+I,\qquad \mathsf{w}_2\mapsto D_2U_1+I,\qquad \mathsf{w}_3\mapsto-D_2U_3+I,$$

and $\mathsf{v}_{i,j}\mapsto d_{ij}u_{ij}+I$ for $i=1,2,3$ and $1<j\leq p_i$. The preceding proposition shows that $\varphi$ is surjective and that the indicated $2\times2$ minors lie in its kernel. The determinantal presentation conjecture. The ring homomorphism $\varphi$ is an isomorphism. This would identify the invariant ring with the quotient of $\mathbb{C}[\mathsf{w},\mathsf{v}]$ by the ideal generated by those $2\times2$ minors; the preceding result establishes the surjectivity and the determinantal relations, while injectivity remains to be shown.