General monotonicity conjecture for plethysm coefficient sequences

Let $\min\mathbb{N}$ and $lin\{0,1,\dotsc,m\}$. For partitions $u$ and $\tau$, let $\mathsf{A}_{l,m}( u,\tau)$ be the sequence of plethysm coefficients defined from the associated skew-shape plethysm coefficients. General plethysm monotonicity conjecture. For all $min\mathbb{N}$, $lin\{0,1,\dotsc,m\}$, and all partitions $u$ and $\tau$, the sequence $\mathsf{A}_{l,m}( u,\tau)$ is weakly increasing. This generalises the Bessenrodt–Bowman–Paget conjecture; the source notes that the cases $l=m$ and $l=0$ are known, while the full assertion remains open.