Invariance of connection matrices under compatible functions

Let $f_1$ and $f_2$ be functions compatible with the same input vector field, and fix the orders of simplices within each Morse set. Connection-matrix invariance conjecture. The connection matrices obtained from $f_1$ and $f_2$ remain the same. This is motivated by the fact that arranging incomparable Morse sets in any order does not change the connection matrix; the source notes that no analogous statement was obtained for Conley-complex persistence.