Interval decomposition conjecture for the nesting-poset zigzag module

Let $P$ be a finite poset, and let $\field P$ denote its incidence algebra. For the critical and regular levels $a_0,t_1,a_1,t_2,\dots,t_n,a_n$ associated with the Morse function $f$, let $\field N_{a_i}$ and $\field N_{t_i}$ be the corresponding incidence algebras of nesting posets. Interval decomposition conjecture. There is a zigzag module of $\field$-algebras

$$\field N_{a_0}\leftrightarrow\field N_{t_1}\leftrightarrow\field N_{a_1}\leftrightarrow\field N_{t_2}\leftrightarrow\cdots\leftrightarrow\field N_{t_n}\leftrightarrow\field N_{a_n},$$

with arrow direction given by Corollary 1, that decomposes into a sum of interval indecomposables of the form

$$\field_I(t)=\begin{cases}\field\{*\}&\text{if }t\in I\\0&\text{if }t\notin I.\end{cases}$$

Furthermore, this collection of interval indecomposables determines $f$ up to poset equivalence. This is presented as a structural persistence claim for the nesting posets; the supplied text gives no evidence of resolution, so it remains open.