Formal identity conjecture for the stratification coefficients

Let $A_{0,1}$ and $A_{s,1}$ be the matrices occurring in the coefficient recursion, put

$$a=\frac{-A_{0,1}}{\beta},$$

and let $\tilde f(T)=\sum_{i=0}^{\infty}a_iT^i\in M_{l\times l}(K)[[T]]$ with $a_0=1$. For $k\geq1$, define $a_k$ inductively by

$$a_k=\frac{1}{k\beta}\left(\sum_{\substack{i+s=k\i\leq k-1}}(d_{i+a,s,1}a_i-A_{s,1})a_i\right).$$

Formal identity conjecture. One has

$$\alpha^a\frac{\tilde f(\alpha t)}{\tilde f(t)}=\sum_{m\geq0}\left(\sum_{n\geq 0}A_{m,n}X_1^{[n]}\right)t^m.$$

Here $\alpha^a$ is defined using the convergence property of $A_{0,1}$. This identity is intended to imply the cocycle condition for the associated stratification.