Reverse stratification conjecture for de Rham prismatic crystals

Let $K$ be the coefficient field, let $M_l(K)$ denote the algebra of $l\times l$ matrices over $K$, and let $\{B_{m,1}\}$ be a sequence in $M_l(K)$. Define matrices $A_{m,n}$ by

$$A_{0,0}=I,\qquad A_{i,0}=0\quad (i>0),\qquad A_{m,1}=B_{m,1},$$

and

$$A_{m,n+1}=(\beta(n-m)+A_{0,1})A_{m,n}+\sum_{\substack{i+j=m\i\leq m-1}}(A_{j,1}+(n-i)\theta_{1,j})A_{i,n}.$$

Assume also that

$$\lim_{n\to+\infty}\prod_{i=0}^n(iE'(\pi)+B_{0,1})=0.$$

Reverse stratification conjecture. The expression

$$\varepsilon(\underline{e})=\underline e\cdot \sum_{m\geq0}\left(\sum_{n\geq 0}A_{m,n}X^{[n]}\right)t^m$$

can serve as the stratification associated to a certain prismatic de Rham crystal; equivalently, it satisfies the cocycle condition.