Explicit modified Derksen invariant for type B varieties

Let $Y$ be a variety, let ${\mathbb K}[Y]$ be its coordinate ring, let $I$ be the ideal generated by the images of all locally nilpotent derivations on $Y$, and let $\mathrm{HD}^*(Y)$ denote the modified Derksen invariant, with $u$ its graded parameter. Modified Derksen invariant conjecture. If $Y$ is of type B, then

$$\mathrm{HD}^*(Y)={\mathbb K}[Y]\oplus\bigoplus_{i>0}Iu^i.$$

The formula would give an explicit description of the modified Derksen invariant in the type B case. The source poses it as an interesting question and does not provide a proof or a resolution.