The WLP conjecture for joins of tangent-space schemes

Let $I_{(1)},\ldots,I_{(\ell)}$ be the tangent-space ideals at $\ell$ general points of the variety of reducible hypersurfaces, let $B$ be the coordinate ring of the join of the schemes $V(I_{(1)}),\ldots,V(I_{(\ell)})$, and let $\mathcal L'$ be the set of general linear forms used for the proper reduction. Write $k=\max\{0,2\ell-n\}$. WLP conjecture. The artinian reduction $B/\mathcal L'B$ has the $k$-weak Lefschetz property. The conjecture is motivated by computer experiments and is used to derive the proposed secant-dimension formula in the improper-intersection range; only various special cases are established in the paper.