The refined symbolic-power containment conjecture with one maximal-ideal factor

Let $K$ be an algebraically closed field, let $M=(x_0,\ldots,x_N)$ be the maximal homogeneous ideal of $K[\mathbb{P}^{N}]$, and let $I$ be the radical ideal of a finite set of points in $\mathbb{P}^{N}$, with $N\geq2$. Refined symbolic-power containment conjecture. For all positive integers $m$ and $t$,

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

This is a proposed refinement of the symbolic-to-ordinary-power containments and is stated in the source as an open question.