Hilbert-function conjecture for minimal local Gorenstein schemes computing cubic cactus rank

Let $F$ be a general cubic form, let $l$ be its local cactus rank, and let a minimal local Artinian Gorenstein scheme computing this rank have Hilbert function $H$. If $l$ is even, then the conjectured Hilbert function is

$$H=(1,l/2-1,l/2-1,1).$$

If $l$ is odd, then it is

$$H=(1,(l-1)/2-1,(l-1)/2-1,1,1).$$

Hilbert-function conjecture. The Hilbert function of a minimal local Artinian Gorenstein scheme for a general cubic form with local cactus rank $l$ has the displayed form according to the parity of $l$. The supplied text presents this as a computation-based conjecture, and gives no resolution or broader range of validity.