Generator conjecture for invariant rings of generalized signatures

Let $n,d\ge 3$, let $V$ be a $d$-dimensional vector space, and let $\sigma:\mathbb{Z}\to\{1,2,\dots,d-1\}$ be a generalized signature of period $n$. Set $V^\sigma=\prod_{j=1}^n\bigwedge^{\sigma(j)}V$ and $R_\sigma=\mathbb{C}[V^\sigma]^{\operatorname{SL}(V)}$. For distinct indices $j_1,\dots,j_k\in[1,n]$, let $u_{j_i}\in\bigwedge^{\sigma(j_i)}V$ be coordinates. Generator conjecture. The invariant ring $R_\sigma$ is generated by the invariants of the form

$$u_{j_1}\curlywedge u_{j_2}\curlywedge\cdots\curlywedge u_{j_k}$$

where

$$\sum_{i=1}^k\sigma(j_i)\equiv0\pmod d.$$

The conjecture gives an explicit finite generating family for the finitely generated factorial invariant ring; the source supplies no resolution.