Steingrímsson's conjecture on partition statistics and q-Stirling numbers

Let ${\cal P}_n^k$ denote the set of partitions of $[n]$ into $k$ blocks, and let $\mathop{\rm mak}$, $\mathop{\rm lmak}$, $\mathop{\rm mak}{}'$ and ${\mathop{\rm lmak}}'$ be the four partition statistics considered in the paper. Let $S_q(n,k)$ denote the classical q-Stirling number of the second kind. Steingrímsson's conjecture. The four statistics have generating function $S_q(n,k)$ on ${\cal P}_n^k$; equivalently,

$$\sum_{\pi\in {\cal P}_n^k}q^{\mathop{\rm mak}(\pi)}=\sum_{\pi\in {\cal P}_n^k}q^{\mathop{\rm mak}{}'(\pi)}=\sum_{\pi\in {\cal P}_n^k}q^{{\mathop{\rm lmak}}'(\pi)}=\sum_{\pi\in {\cal P}_n^k}q^{\mathop{\rm lmak}(\pi)}=S_q(n,k).$$

The conjecture concerns q-analogues of Stirling-number generating functions for set partitions; the supplied text identifies it as Steingrímsson's conjecture but gives no resolution status.