Suppose we have a submartingale $X=\{X_t\}_t$ and a supermartingale $Y=\{Y_t\}_t$ which are adapted to the same filtration on a bounded set and have a common initial value $X_0=Y_0$. Suppose that $\tau^*_p$ is a stopping time that solves $$ \max_{\tau}\ p\, \mathbb E[\delta^\tau X_{\tau}]+(1-p)\,\mathbb E[\delta^\tau Y_{\tau}] $$ where $\delta\in(0,1)$ and maximization is over all stopping times that are measurable with respect to the filtration generated by the processes $X$ and $Y$. I want to prove or disprove that $$q>p\Rightarrow \tau^*_q\ge \tau^*_p$$
Intuitively, suppose an investor owns an asset and is uncertain about the process that governs its future value. He believes that with probability $p$ the value of the asset is governed by $X$ and with probability $1-p$ it is governed by $Y$. The statement I am trying to prove is that the investor will wait longer the more optimistic he is that the value will increase over time.
Any hints are greatly appreciated!