The Eisenbud–Green–Harris conjecture for complete intersections

Let $A=B=K[X_1,\dots,X_n]$, let $\mathfrak a=(f_1,\dots,f_r)$ be generated by a homogeneous regular sequence with degrees $d_1\leq\cdots\leq d_r$, and let $R=A/\mathfrak a$. Set $\mathfrak b=(X_1^{d_1},\dots,X_r^{d_r})$ and $S=B/\mathfrak b$. Let $\epsilon_{\mathrm{CL}}$ be the Clements–Lindström embedding of the lex-segment ideals of $B$ into $S$. Eisenbud–Green–Harris conjecture. Let $\mathbf f=(f_1,\dots,f_r)$ be as above. Then $R$ embeds into $(S,\epsilon_{\mathrm{CL}})$. Few cases are known, including cases covered by results of Clements and Lindström, Cartwright and Madsen, Chen, and Ab. The general assertion remains open.