Minimizer of Loss Function

Question

I am trying to solve for the minimizer of $L_{\tau}(y,\hat{y}):=E[l_{\tau}(y,\hat{y})|x]$, where $l_{\tau}=(y-\hat{y})(\tau−1_{(y−\hat{y}<0)})$. Assume that $0\leq\tau\leq1$. I have plotted $y$ and $\hat{y}$ for different value of $\tau$ and am pretty confident that the answer is $E[y|x]$. However, I got stuck on where to start. Any help is appreciated!