Algebraization conjecture for 6-webs with cubic-threefold properties

Let $\boldsymbol{\mathcal W}$ be a 6-web satisfying the five listed properties: it is linearizable and skew, has maximal 2-rank equal to $10$, has 1-rank at least $5$, and has a 5-dimensional subspace $\boldsymbol{A}^1(\boldsymbol{\mathcal W})\subset\boldsymbol{AR}^{(1)}(\boldsymbol{\mathcal W})$ whose wedge map to $\boldsymbol{AR}^{(2)}(\boldsymbol{\mathcal W})$ is an isomorphism. Algebraization conjecture. The web $\boldsymbol{\mathcal W}$ is equivalent to an algebraic web $\boldsymbol{\mathcal L\hspace{-0.05cm} W}_X$ associated with a cubic hypersurface $X\subset\mathbf P^4$. This conjecture proposes a characterization of the webs arising from cubic hypersurfaces, in analogy with the algebraization result cited in the source; no resolution is given there.