Let $X_{AB}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$. Suppose that $X_{AB}$ is block positive, meaning that (in Dirac notation)

$\langle \psi |_A \otimes \langle \varphi|_B X_{AB} | \psi \rangle_A \otimes | \varphi \rangle_B \geq 0 $

for all vectors $| \psi \rangle_A \in \mathcal{H}_A$ and $| \varphi \rangle_B \in \mathcal{H}_B$. Let us abbreviate this as

$X_{AB} \geq_{\text{BP}(A:B)} 0$.

Now let $Y_{CD}$ be an operator acting on the tensor-product Hilbert space $\mathcal{H}_C \otimes \mathcal{H}_D$. Suppose that

$Y_{CD} \geq_{\text{BP}(C:D)} 0.$

Is it then the case that

$X_{AB} \otimes Y_{CD} \geq_{\text{BP}(AC:BD)} 0$

or is there a counterexample?