The composition-lift conjecture for path-complete Lyapunov functions

Let $d4a2,d4a2$ be path-complete Lyapunov functions in $d52f\mathrm{pc}(\Sigma)$ satisfying Assumption~. Suppose that $d53d$ is a subset of linear invertible maps. The composition-lift conjecture. The following statements are equivalent:

1. $d4a2^{\circ}$ simulates $d4a2$.
2. $d4a2\leq_{\mathcal{V},\mathcal{F}}d4a2$ for every template $d4a2$ closed under forward composition with the class of dynamics $d53d$.

This conjecture concerns whether simulation by the composition lift is exactly characterized by the corresponding preorder for every forward-composition-closed template. It was expected in work on composition lifts and is identified in the source with Conjecture 8.20 of Debauche (2024); the supplied material does not establish its resolution.