Iarrobino's conjecture on powers of generic linear forms

Let $f_1,\ldots,f_r$ be generic forms of degree $d$ in $k[x_1,\ldots,x_n]$, and let $l_1,\ldots,l_r$ be generic linear forms. Exclude the cases $r=n+2$, $r=n+3$, $(n,r)=(3,7),(3,8),(4,9),(5,14)$.

Iarrobino's conjecture. Outside these exceptions, the algebras

$$k[x_1,\ldots,x_n]/(l_1^d,\ldots,l_r^d)\quad\text{and}\quad k[x_1,\ldots,x_n]/(f_1,\ldots,f_r)$$

have the same Hilbert series.

The conjecture compares powers of generic linear forms with generic forms of the same degree. The source does not specify its resolution status.