Restriction conjecture for cubic forms of slice rank at most r

Let $V$ be a finite-dimensional vector space, let $f\in S^3V$, and let $E\subseteq V^*$ be a linear subspace. Write $f|_E$ for the restriction of $f$ to $E$, and let $\mathrm{slrk}(f)$ denote the slice rank of $f$. Restriction conjecture. A system of set-theoretic equations for $\sigma_r(\mathcal{R}^1_{n,3})$ arises from restrictions to $2r+1$ variables. More precisely, the following statements are equivalent:

$$\begin{array}{ll} \mathrm{(i)} & \mathrm{slrk}(f)\leq r;\\ \mathrm{(ii)} & \text{for a generic }E\subseteq V^*\text{ with }\dim E\leq 2r+1,\ \mathrm{slrk}(f|_E)\leq r;\\ \mathrm{(iii)} & \text{for every }E\subseteq V^*\text{ with }\dim E\leq 2r+1,\ \mathrm{slrk}(f|_E)\leq r. \end{array}$$

This would reduce the construction of set-theoretic equations for the secant variety $\sigma_r(\mathcal{R}^1_{n,3})$ to restrictions to $2r+1$ variables, extending the analogous low-dimensional restriction results discussed in the paper. The source does not provide a resolution of the conjecture.