AMC 10 2022b Problem 13

The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes? $\textbf{(A)}\ 8 \textbf{(B)}\ 10 \textbf{(C)}\ 11 \textbf{(D)}\ 13 \textbf{(E)}\ 16$