Steingrímsson's Euler–Mahonian conjecture for ordered partitions

Let $\mathcal{OP}_n^k$ be the set of ordered partitions of $[n]$ into $k$ blocks, and let $\operatorname{mak}$, $\operatorname{lmak}$, $\operatorname{bInv}$, and $\operatorname{bMaj}$ be the statistics defined in the paper. Set

$$\operatorname{cbInv}=\binom{k}{2}-\operatorname{bInv},\qquad \operatorname{cbMaj}=\binom{k}{2}-\operatorname{bMaj},$$

and define

$$\operatorname{cinvLSB}=\operatorname{lsb}+\operatorname{cbInv}+\binom{k}{2},\qquad \operatorname{cmajLSB}=\operatorname{lsb}+\operatorname{cbMaj}+\binom{k}{2}.$$

Steingrímsson's conjecture. The statistics

$$\operatorname{mak}+\operatorname{bInv},\quad \operatorname{lmak}+\operatorname{bInv},\quad \operatorname{mak}+\operatorname{bMaj},\quad \operatorname{lmak}+\operatorname{bMaj},$$

and $\operatorname{cinvLSB}$ and $\operatorname{cmajLSB}$ are Euler–Mahonian on $\mathcal{OP}_n^k$: their generating functions over $\mathcal{OP}_n^k$ are all equal to

$$[k]_q!S_q(n,k).$$

This conjecture concerns equidistribution of partition statistics and their $q$-Stirling generating functions; the source presents it as the reduction of conjectures from Steingrímsson. The supplied material does not establish whether it has been resolved.