I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and have (for simplicity) two stopping times $\tau_1$ and $\tau_2$ such that $\tau_2 \leq \tau_1$ and $U:\Bbb R\rightarrow \Bbb R$ is a continuous function which is concave over $\Bbb R^+$ and convex otherwise. My aim is to show that:

$$\sup_{\tau_2\leq\tau_1}\Bbb E\left[ U \left( \sum_{i=1}^2Y_{\tau_i} \right) \mid F_0 \right] = \sup_{\tau_2}\Bbb E \left[ \sup_{\tau_1} \Bbb E \left[ U \left( \sum_{i=1}^2 Y_{\tau_i}\right)\mid F_{\tau_2}\right] \mid F_0\right] $$

I understand that part of it is simply the application of the "tower property" but not really sure how to go around the exchange between the supremum and the Expectation.

Any help would be greatly appreciated. Thanks in advance :)