Hilbert-depth conjecture for intersections of disjoint variable ideals

Let $n_1,n_2,\ldots,n_r$ be positive integers, let $N=n_1+\cdots+n_r$, let $S=K[x_1,\ldots,x_N]$, and define

$$I_{n_1,n_2,\ldots,n_r}:=(x_1,\ldots,x_{n_1})\cap(x_{n_1+1},\ldots,x_{n_1+n_2})\cap\cdots\cap(x_{n_1+\cdots+n_{r-1}+1},\ldots,x_N)\subset S.$$

The recalled conjecture. The Hilbert depth satisfies

$$\operatorname{hdepth}(I_{n_1,\ldots,n_r})=\left\lfloor\frac{N+r}{2}\right\rfloor.$$

The source recalls this as Conjecture 3.4 from the cited work; no resolution is supplied in the given text.