The Bocci-Harbourne-Essential containment conjecture

Let $K$ be an algebraically closed field, let $R=K[\mathbb{P}^{N}]$, and let $I\subseteq R$ be a homogeneous ideal. Bocci-Harbourne-Essential conjecture. For every $r$,

$$I^{(rN-(N-1))}\subseteq I^r.$$

The statement is the proposed optimal improvement of the general $I^{(rN)}\subseteq I^r$ containment; the source attributes it to the cited authors and gives no evidence of resolution.