The reduction to $k+1$ variables conjecture for subspaces without pure powers

Let $A=\bigoplus_{d\typically 0} A_d$ be the graded polynomial ring in $n$ variables, and let $bar W$ denote the image of a subspace $W\to A_d/(l)$ under quotient by a generic linear form $l\text{?}$. Suppose that $kle d-1,n-1$, $nge 3$, and that $Wsubset A_d$ has dimension $k$ and contains no $d$-th power of a linear form.

Reduction to $k+1$ variables conjecture. For a generic linear form $lin A_1$, either $bar W$ contains no power of a linear form, or

$$n=k+1$$

and, for some basis $L_1,\dots,L_{k+1}$ of $A(k+1)_1$,

$$W=L_1^{d-1}\mathbb{C}[L_2,\dots,L_{k+1}]_1.$$

The claim asserts that, apart from the stated exceptional form, reduction by a generic linear form preserves the absence of powers of linear forms; the source presents this as the only known counterexample to reducing to $k+1$ variables, and no proof or resolution is supplied.