Proving that $\{\mathbb G_n(t)/t, \mathcal F_n(t)\}$ is a reverse martingale (tips for proving reverse martingales)

Question

We want to prove that $\{\mathbb G_n(t)/t, \mathcal F_n(t), 0 < t \leq 1\}$ is a reverse martingale, where $n\mathbb G_n(t) = \#\{i \in \{1, \ldots, n\} : \xi_i \leq t\}$ where $\xi_i$ are all i.i.d. Uniform$(0,1)$, and where $\mathcal F_n(t) = \sigma[\mathbb G_n(u) : t \leq u \leq 1]. $ In other words, we want to prove that $$\mathbb E\left(\frac{\mathbb G_n(s)}{s} \bigg|\mathcal F_n(t) \right) = \frac{\mathbb G_n(t)}{t}$$ My main problem is that I don't really know how to work with reverse martingales; if it were a normal martingale I would just use some independence tricks to pull things out of the conditional expectation; however I don't know what to do here. I was thinking we could prove $$\mathbb E\left(n\mathbb G_n(s) \Big|\mathcal F_n(t) \right) = n\mathbb G_n(t) \frac{s}{t}$$ (which makes sense intuitively?) but again I'm not really sure how to get there mathematically.