Equidistribution of minimaj with inversion, major index, and diagonal inversion

Given an ordered set partition $\pi\in\mathcal{OP}_n$, define its shape $\operatorname{shape}(\pi)$ to be the composition whose $i$th part is the size of the $i$th block of $\pi$ from left to right. Let $D^{\operatorname{stat}}_{\alpha,k}(q)$ denote the corresponding statistic-generating polynomial for ordered multiset partitions of content $\alpha$ with $k$ blocks. Minimaj equidistribution conjecture. For every composition $\beta$ of length $n$,

$$\sum_{\substack{\pi\in\mathcal{OP}_n\operatorname{shape}(\pi)=\beta}}q^{\operatorname{inv}(\pi)} = \sum_{\substack{\pi\in\mathcal{OP}_n\operatorname{shape}(\pi)=\beta}}q^{\operatorname{minimaj}(\pi)}.$$

Moreover, for every composition $\alpha$,

$$D^{\operatorname{minimaj}}_{\alpha,k}(q)=D^{\operatorname{inv}}_{\alpha,k}(q)=D^{\operatorname{maj}}_{\alpha,k}(q)=D^{\operatorname{dinv}}_{\alpha,k}(q).$$

The two assertions are stated separately, and neither directly implies the other. They are motivated by computational data suggesting that $\operatorname{minimaj}$ has the same distribution as inversion, major index, and diagonal inversion, but the authors do not have a proof with their available techniques.