A quadratic polynomial with real coefficients and leading coefficient $\frac{1}{2}$ is called disrespectful if the equation $p(p(x))=0$ is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial $\tilde{p}(x)$ for which the sum of the roots is maximized. What is $\tilde{p}(2)$?