A positive-semidefinite minimizing eigenmatrix

For real symmetric positive definite $n\times n$ matrices $A$ and $B$, the operator $\mathcal L(X)=AXB+BXA$, whose matrix under column vectorization is $A\otimes B+B\otimes A$, has a nonzero real positive-semidefinite eigenmatrix for its smallest eigenvalue. The inverse of $\mathcal L$ is written as $\int_0^\infty M_tYM_t\,dt$ with real symmetric $M_t$, so it preserves positive semidefiniteness. For a symmetric or skew-symmetric eigenmatrix $W$ belonging to the largest eigenvalue of this inverse, the modulus $|H|$ of $H=W$ or $H=iW$ is real, symmetric and positive semidefinite, and its Rayleigh quotient is at least as large, so it is also an eigenmatrix. Consequently, the minimum Rayleigh quotient of $A\otimes B$ over vectorized symmetric matrices is at most its minimum over skew-symmetric ones, without commutativity, rank or simple-eigenvalue assumptions.