Watanabe's bold conjecture on embeddings into quadratic complete intersections

Let $\mathbb{F}$ be a field and let $R=\mathbb{F}[x_1,\ldots,x_n]$ have its standard grading. A complete intersection algebra is a quotient $R/\langle f_1,\ldots,f_n\rangle$ by a homogeneous regular sequence, and a quadratic complete intersection algebra is one whose generators are all quadratic forms. A graded algebra homomorphism between complete intersections is an embedding when it maps a socle generator onto a socle generator. Watanabe's bold conjecture. Every complete intersection algebra embeds into some quadratic complete intersection algebra. The embedding can occur only between complete intersections with the same socle degree. The conjecture concerns the extent to which complete intersections arise from quadratic complete intersections; the paper verifies it when the defining polynomials split into linear factors, while the general statement is not resolved here.