The LMAX extension of the set-valued Wilf-equivalence problem

Let $P$ be a set of indecomposable patterns. Let $\mathsf{LMAX}$ denote the set-valued statistic recording left-to-right maxima, and let $\mathsf{iar}$ and $\mathsf{comp}$ denote the initial ascending run length and number of components, respectively. LMAX extension conjecture. For every $n$,

$$\left|\mathfrak{S}_n(P)^{\mathsf{LMAX},\mathsf{iar}}\right|=\left|\mathfrak{S}_n(P)^{\mathsf{LMAX},\mathsf{comp}}\right| \Longleftrightarrow \left|\mathfrak{S}_n(P)^{\mathsf{LMAX},\mathsf{iar},\mathsf{comp}}\right|=\left|\mathfrak{S}_n(P)^{\mathsf{LMAX},\mathsf{comp},\mathsf{iar}}\right|.$$

The paper presents this as a suspected special case of an open problem concerning any totally $\oplus$-compatible set-valued statistic, and notes that it is expected to hold for $\mathsf{LMAX}$; no proof or resolution is supplied.