The algebraic packing problem for squarefree monomial ideals

Let $S$ be a polynomial ring and let $I$ be a squarefree monomial ideal. A minor of $I$ is obtained by setting variables successively equal to $0$ or $1$ in its minimal monomial generators and taking the ideal generated by the resulting monomials. The ideal has the packing property when every minor satisfies the König property, equivalently when its height equals its monomial grade. Let $I^{(n)}$ denote the $n$th symbolic power.

Algebraic packing conjecture. One has

$$I^n=I^{(n)}\quad\text{for all }n\geq 1$$

if and only if $I$ has the packing property.

This is presented as the commutative-algebra translation of the packing problem; the supplied text does not give a resolution.