Quadratic form of Kroenecker products of skew-symmetric matrices

Question

I am trying to understand under which conditions on $P=P^\top>0$ , $C=C^\top$, the following quadratic form is zero:

$$ x^\top \left( D U^\top \frac{L-L^\top}{2} U \otimes PC \right)x = 0 $$

where $D$ is diagonal, $U$ is unitary, $\frac{L-L^\top}{2}$ is a skew-symmetric matrix. If needed, $L$ has also zero row-sum (is a Laplacian matrix). Thanks in advance!!