The powers-of-generic-forms conjecture for equal degrees

Let $R=\mathbb{C}[x_1,\ldots,x_n]$, with $g_1,\ldots,g_k$ generic forms of degree $d\geq2$, and let $m\in\mathbb{N}$. Set

$$I=(g_1^m,g_2^m,\ldots,g_k^m).$$

For a power series $F(t)$, write $\lceil F(t)\rceil$ for its truncation at the first nonpositive coefficient. The powers-of-generic-forms conjecture.

$$H_{R/I}(t)=\left\lceil\frac{(1-t^{md})^k}{(1-t)^n}\right\rceil.$$

This asserts that powers of generic degree-$d$ forms have the Hilbert series predicted for $k$ generic forms of degree $md$. The source presents this as motivated by computational simplicity and does not report a resolution.