Euler–Mahonian conjecture for the LSB statistics on ordered partitions

Let $[n]$ be partitioned into $k$ ordered blocks. Let $\lsb$ be the left-smaller-block statistic, and let \mathop{\text{\text{\rm cb}\text{\maj}}} and \mathop{\text{\text{\rm cb}\text{\inv}}} denote the corresponding block major-index and block inversion statistics. Define

$$\mathop{\text{\text{\rm cmaj}\text{$\LSB$}}}=\lsb+\mathop{\text{\text{\rm cb}\text{$\maj$}}}+\binom{k}{2},\qquad \mathop{\text{\text{\rm cinv}\text{$\LSB$}}}=\lsb+\mathop{\text{\text{\rm cb}\text{$\inv$}}}+\binom{k}{2}.$$

LSB Euler–Mahonian conjecture. The statistics \mathop{\text{\text{\rm cmaj}\text{\LSB}}} and \mathop{\text{\text{\rm cinv}\text{\LSB}}} are Euler–Mahonian on ordered partitions of $[n]$ into $k$ blocks. On ordinary partitions, both reduce to $\lsb+\binom{k}{2}$, which is already known to be Euler–Mahonian; the ordered-partition assertion remains conjectural.