The iterated Weak Lefschetz Property conjecture for quadratic complete intersections

Let $k$ be a field of characteristic zero and let

$$R=k[x_1,\dots,x_n]/(x_1^2,\dots,x_n^2).$$

For linear forms $\ell_1,\dots,\ell_i$ in $R$, set

$$R_i=R/(\ell_1,\dots,\ell_i).$$

Iterated Weak Lefschetz Property conjecture. For any linear forms $\ell_1,\dots,\ell_i$ in $R$, the quotient $R_i$ enjoys the Weak Lefschetz Property.

The paper has already established that $R$ itself enjoys the Weak Lefschetz Property in characteristic zero, but the corresponding assertion for every iterated quotient is presented here without a supplied resolution.