Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$.

Show that
$$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$

This question has been asked on MSE without receiving any answers. Just some comment by achille hui and myself.

I think this result is true because for $n=1$ we have $6(A^3+B^3+C^3)+1\ge 5(A^2+B^2+C^2)$

but unfortunately I didn't succeed to generalize for $n\ge2$.

N.B.: My apologize if this question is not suitable on math overflow.