# powers of linear forms in projections of complete intersections in codimension 3

Let $$I\subset \mathbb{C}[x_0,x_1,x_2]=:A$$ be a complete intersection, $$I=(p_1,p_2,p_3)$$, $$p_i$$ homogeneous all of the same degree d for some $$d>2$$.

Let $$l$$ be a general linear form and let $$J$$ be the image of $$I$$ in $$A/(l)$$. Then $$J$$ is still minimally generated by three elements of degree $$d$$ by Green's hyperplane restriction theorem.

Is it true that if $$I$$ contains no $$d$$-th power of a linear form, then the same holds for $$J$$ (in $$A/(l)$$)?