Let $\mathbb{S}^n_+$ denote the space of $n \times n$ symmetric positive semidefinite matrices. Let $A \in \mathbb{S}^n_+$ and $B \in \mathbb{S}^n_+$. Then $\langle A, B\rangle_F = \text{Tr}(A^\top B) = \text{vec}(A)^\top \text{vec}(B) \geq 0$. Let $K \in \mathbb{S}_+^{n^2}$. Is it true that $$ \text{vec}(A)^\top K \, \text{vec}(B) \geq 0 ?$$
Special case: I believe the answer would be true if $K = \matrix C \otimes C$ for some matrix $C$, since $K \text{vec}(B) = \text{vec}(C^\top B C)$ and the desired inner product is just the inner product of two PSD matrices. How about more generally?