Motivated by this question I propose a simplification:
Question Let matrices $A,B,C\in M_{2}(\mathbb{C})$ be Hermitian and positive definite, such that:$$A+B+C=I_2$$ Show that: $$\det\left(6(A^3+B^3+C^3)+I_{2}\right)\ge 5^2\det(A^2+B^2+C^2)$$ where $I_{2}$ is the identity matrix.
The original question is of unknown origin and I am hoping for a substantial simplification in the $2 \times 2$ case where the following expansion holds:
$$\left[ \begin{array}{cc} x_1 & y + iz \\ y - iz & x_2\end{array}\right] = \left[ \begin{array}{cc} x_1 & 0 \\ 0 & x_2\end{array}\right] + y\left[ \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array}\right] + z\left[ \begin{array}{rc} 0 & i \\ - i & 0\end{array}\right]$$
maybe with expansion to Pauli spin matrices this is solvable.
There is a nice article by Knutson and Tao on Honeycombs that might be of assistnce: