Of course I know these things. But (\mathop{\textrm{diag}}\alpha)-\alpha\alpha^T is a nonlinear expression. There is no guarantee that log-concavity will be preserved if you give the determinant function a nonlinear argument.
This is basic convex analysis. It is the reason why the composition rules are such a key part of the DCP ruleset, and why you must read and obey the FAQ if you intend to ask for help here.