Let $A$ be a (symmetric) positive semidefinite $n \times n$ matrix with diagonal $D$.
Let $k \in \{1, \dots, n\}$ and $M = A - k\cdot D$. Prove that the sum of the top $n-k+1$ eigenvalues of $M$ is nonpositive.
For example, when $k = 1$, the trace of $M$ is $0$, and when $k = n$, all the eigenvalues of $M$ are nonpositive (this can be seen by Cauchy-Schwarz).
We can reduce it to the case when $A$ has rank 1 by convexity, and in that case we can even write down the characteristic polynomial by using the matrix determinant lemma. It doesn't seem to make the problem easier.