Relation between continuous and discrete Rayleigh systems

Consider a continuous Rayleigh system $(L,\mathcal{R})$ on $TQ$, where $L$ is natural, meaning that

$$L(q,\dot{q})=\frac{1}{2}g(\dot{q},\dot{q})-V(q).$$

Here $g$ is a Riemannian metric on $Q$ and $V$ is a function on $Q$. Let $f_L=-S^*(\mathrm{d}\mathcal{R})$ be the continuous Rayleigh force, and let $f_d^E$ be the exact discrete force on $Q\times Q$ associated with $f_L$. Relation between continuous and discrete Rayleigh systems. The exact discrete force $f_d^E$ is Rayleigh. The claim is motivated by computations in several examples, but no proof is provided; its status is therefore unresolved.