Orbit-harmonics containment conjecture for rectangular set partitions

Let $Pi_{(b^a)}$ and $Pi_{(a^b)}$ denote the loci of unordered set partitions whose block-size types are $(b^a)$ and $(a^b)$, respectively, and let ${\rm gr}{\bf I}( Pi)$ be the associated graded vanishing ideal in the polynomial ring on the variables indexed by $\binom{[n]}{2}$. Orbit-harmonics containment conjecture. For positive integers $a\ge b$, one has

$${\rm gr}{\bf I}(\Pi_{(b^a)})\subseteq{\rm gr}{\bf I}(\Pi_{(a^b)}).$$

This is presented as a stronger conjecture related to Foulkes' conjecture, since the orbit-harmonics construction gives ${\rm Frob}(R(\Pi_{(b^a)}))=h_a[h_b]$.