Well-presented generators conjecture for joins of secant varieties and coordinate spaces

Let $\mathcal I(\boldsymbol J(\sigma_{r,n},L^S))$ be the ideal of the join of the relevant secant variety $\sigma_{r,n}$ and coordinate linear space $L^S$. A generator is called well presented when its propagated polynomial, constructed as in the source, is a generator of the corresponding ideal after simultaneously increasing the indices by every $q$. Well-presented generators conjecture. For all $r,n,S$, there exists a set of generators $P^1,\ldots,P^{\mu}$ of

$$\mathcal I(\boldsymbol J(\sigma_{r,n},L^S))$$

that can be well presented. This conjecture seeks generators compatible with the propagation operation across all enlarged instances. The source gives no resolution of the conjecture.