AMC 10 2021-fall-a Problem 25

A quadratic polynomial with real coefficients and leading coefficient $1$ is called $\emph{disrespectful}$ 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 $\tilde{p}(1)$? $\textbf{(A) } \dfrac{5}{16} \textbf{(B) } \dfrac{1}{2} \textbf{(C) } \dfrac{5}{8} \textbf{(D) } 1 \textbf{(E) } \dfrac{9}{8}$