Primary decomposition conjecture for powers of star-configuration ideals

Let $s>n$ and let $\mathcal H=\{H_1,\ldots,H_s\}$ be properly meeting hyperplanes in ${\bf P}^n$, defined by linear forms $L_i$. Let $N+1=s$, let $\mathcal H'$ be the coordinate hyperplanes in ${\bf P}^N$, and define $\phi:k[{\bf P}^N]\to k[{\bf P}^n]$ by $\phi(x_i)=L_{i+1}$. Let $M$ and $M'$ be the irrelevant ideals in the respective coordinate rings. Primary decomposition conjecture. For every relevant choice of $c$ and $l$, the inclusions in the displayed chain relating $I_{V_c(\mathcal H,{\bf P}^n)}^l$ to the intersections of the images under $\phi$ are equalities; equivalently, the first inclusion asserts that $\phi$ commutes with the indicated intersections, and the last inclusion asserts that the irrelevant component can be replaced by $M^{(N-c+2)l}$. This would give a primary decomposition of $I_{V_c(\mathcal H,{\bf P}^n)}^l$ in terms of the indicated symbolic powers, with the tail component primary for the irrelevant ideal $M$.