Fix an integer $k>4$. For any integer $r>0$, denote by $S_{r}:=\mathbb{C}[X_0,X_1,X_2,X_3]_{r}$ the vector space of degree $r$ polynomials in $X_i$ with coefficients in $\mathbb{C}$. Let $W$ be a vector subspace of $S_{2k}$ of codimension $1$ such that $W_{2k+1}=S_{2k+1}$ where $W_{2k+1}$ is the degree $2k+1$ part of the ideal (of $\mathbb{C}[X_0,X_1,X_2,X_3]$) generated by $W$. Define an ideal $I$ by $I_r$ is the biggest vector space in $S_r$ such that $I_r \otimes S_{2k-r} \subset W$ for $0< r<2k$. This ideal is Gorenstein of socle degree $k$.
The question is when is this ideal generated in degree less than or equal to $k$ i.e., we can find a set of generators of $I$ such that the degrees of the generators is less than or equal to $k$? Is it true that the codimension of $I_r$ in $S_r$ is a strictly increasing function for $r \le k$?
A good reference for this topic will be great help as well.
Edit: In the first para we add an extra assumption that the induced pairing $S_j/W_j \times S_{2k-j}/W_{2k-j} \to S_{2k}/W_{2k} \cong \mathbb{C}$ is a perfect pairing for $j<2k$.
