The two-part refined symbolic-power containment conjecture

Let $K$ be an algebraically closed field, let $I$ be the radical ideal of a finite set of points in $\mathbb{P}^{N}$, and let $M=(x_0,\ldots,x_N)$ be the maximal homogeneous ideal of $K[\mathbb{P}^{N}]$. Refined two-part containment conjecture. For all positive integers $m$ and $t$, both containments hold:

$$I^{(t(m+N-1)-N+1)}\subseteq (I^{(m)})^t$$

and

$$I^{(t(m+N-1)-N+1)}\subseteq M^{(t-1)(N-1)}(I^{(m)})^t.$$

The source explains that this conjecture simultaneously implies earlier containment conjectures and presents it as an unresolved synthesis of them.