Conjecture on simultaneous analytic images of Liouville matrices

Let $n\ge 2$ and let $I\subseteq\mathbb{R}$ be an open interval containing $0$. Let $\mathscr{L}_{n,n}$ denote the set of Liouville matrices, and let $\textswab{f}_k$ and $\textswab{g}_k$ be the extensions associated with sequences of analytic functions $f_k,g_k:I\to\mathbb{R}$, respectively. For $a,b,c,d\in\mathbb{R}$, let $\tau_{a,b,c,d}$ denote the function referred to in the source as equation (4). Assume that, for every $k\ge 1$, $g_k$ is not of the form $\tau_{a,b,c,d}$, and that no $f_{k_1}$ is identical on $I$ to $\tau_{a,b,c,d}\circ g_{k_2}$ for any $a,b,c,d\in\mathbb{R}$ and $k_1,k_2\in\mathbb{N}$. Conjecture on simultaneous analytic images of Liouville matrices. There exists $A\in\mathscr{L}_{n,n}$ such that

$$\textswab{f}_k(A)\in\mathscr{L}_{n,n}\quad\text{and}\quad \textswab{g}_k(A)\notin\mathscr{L}_{n,n}$$

for every $k\ge 1$. Equivalently,

$$\bigcap_{k\ge 1}\left(\textswab{f}_k^{-1}(\mathscr{L}_{n,n})\cap\textswab{g}_k^{-1}(\mathscr{L}_{n,n}^{c})\right)\cap\mathscr{L}_{n,n}\ne\emptyset.$$

This conjecture combines the preceding results on analytic images of Liouville matrices and would provide a simultaneous positive/negative image theorem under the stated exclusions. The supplied context does not indicate whether it has been resolved.