The conjecture on determinants of fundamental matrices for reflected systems

Consider the differential system

$$Fu'(t)+Gu'(-t)+Au(t)+Bu(-t)=0,$$

for $t\in\mathbb R$, where $n\geq 1$, $A,B,F,G\in\mathcal M_n(\mathbb R)$, and $u:\mathbb R\to\mathbb R^n$. Assume that $F-G$ and $F+G$ are invertible, and let $X(t)$ be a fundamental matrix. Define

$$E=(F-G)^{-1}(A-B)(F+G)^{-1}(A+B).$$

Determinant conjecture. The determinant $|X(t)|$ can be obtained as a component of the solution of a linear system of differential equations with constant coefficients, whose coefficients depend only on the different matrix invariants of $E$.

This conjecture seeks a higher-dimensional analogue of the Abel–Jacobi–Liouville identity, which gives such a constant-coefficient system when $n=2$.