The generic linear-form reduction conjecture for subspaces without powers

Let $A={\mathbb{C}}[x_1,\dots,x_n]$ be a polynomial ring, let $A_d$ denote its degree-$d$ component, and let $W\subseteq A_d$ be a $k$-dimensional subspace with $k\le d-1$ and $k\le n-1$, where $n\ge3$. Write $\overline W$ for the image of $W$ after the relevant generic linear-form reduction. Generic linear-form reduction conjecture. If $W$ contains no $d$-th power of a linear form, then for a generic linear form $l\in A_1$, either $\overline W$ contains no $d$-th power of a linear form, or $n=k+1$ and $W=L_1^{d-1}{\mathbb{C}}[L_2,\dots,L_{k+1}]_1$ for some basis $L_1,\dots,L_{k+1}$ of $A(k+1)_1$. This proposed reduction would sharpen the preceding reduction to $2k$ variables to $k+1$ variables; the surrounding text presents it as likely and notes that only one counterexample is known, with small monomial cases checked computationally.