Steingrímsson's Euler–Mahonian conjecture for ordered set partitions

Let ${\cal OP}_n^k$ be the set of ordered $k$-partitions of $[n]$, and let $\mathop{\rm bmaj}$ and $\mathop{\rm binv}$ be the block-major index and block-inversion statistics. Let $\mathop{\rm mak}$, $\mathop{\rm mak}{}'$, $\mathop{\rm lmak}{}'$ and $\mathop{\rm lmak}$ be the four statistics from the preceding definition. A statistic on ${\cal OP}_n^k$ is Euler–Mahonian when its generating function is $[k]_q!S_q(n,k)$. Steingrímsson's conjecture. Each of the following eight statistics is Euler–Mahonian:

$$\mathop{\rm mak}+\mathop{\rm bmaj},\quad \mathop{\rm mak}{}'+\mathop{\rm bmaj},\quad \mathop{\rm lmak}{}'+\mathop{\rm bmaj},\quad \mathop{\rm lmak}+\mathop{\rm bmaj},$$ $$\mathop{\rm mak}+\mathop{\rm binv},\quad \mathop{\rm mak}{}'+\mathop{\rm binv},\quad \mathop{\rm lmak}{}'+\mathop{\rm binv},\quad \mathop{\rm lmak}+\mathop{\rm binv}.$$

The claim proposes Euler–Mahonian extensions of q-Stirling statistics to ordered set partitions; the supplied text gives no evidence that it has been proved or disproved.