Let $A$ and $B$ be two real matrices such that $AB$ is positive semi-definite where positive semi-definite is to mean that $x^TMx \geq 0$ for all $x$.
Let $D$ be a diagonal, full rank, square, positive definite, real matrix (i.e has only positive values along the diagonal).
Is $ADB$ positive semi-definite?
If not - are there any conditions on non-square $A$ and $B$ that do make $ADB$ positive semi-definite?
From Theorem 2 in https://cms.math.ca/openaccess/cjm/v15/cjm1963v15.0313-0317.pdf, we know that the product of three positive definite matrices is positive definite if the product is Hermitian. But I wasn't sure if that extended to this use case.