Nicklasson's conjecture on powers of generic forms

Let $S$ be a polynomial ring over $\mathbb C$, and let $g_1,\ldots,g_r$ be generic forms of degree $d>1$. Nicklasson's conjecture. The ideal

$$(g_1^k,\ldots,g_r^k)$$

has the same Hilbert series as an ideal generated by $r$ generic forms of degree $dk$. This compares powers of generic forms with generic forms of the resulting degree; it is known in the binary-form case and remains open in general.