Let $H,K$ be complex Hilbert spaces and $\operatorname{HS}(H),\operatorname{HS}(K)$ the spaces of Hilbert-Schmidt operators. I will identify $\operatorname{HS}(H)$ with $H\otimes \overline{H}$ (where $\overline{H}$ is the conjugate Hilbert space, $\overline{H}=\{\overline{\xi} | \xi\in H\}$) and $\operatorname{HS}(H)\otimes\operatorname{HS}(K)$ with $\operatorname{HS}(H\otimes K)$. Denote by $\top$ the canonical map $\operatorname{B}(H)\rightarrow \operatorname{B}(\overline{H})$ and by $J\colon \operatorname{HS}(H\otimes K)\rightarrow \operatorname{HS}(H\otimes K)$ the adjoint map $J(T)=T^*$, it is antilinear isometry.
Fix an operator $T=\sum_{n=1}^{N} A_n\otimes \mathbb{1}_{\overline{H}}\otimes \mathbb{1}_{K} \otimes B_n^{\top}$ with $A_n\in\operatorname{B}(H),B_n\in \operatorname{B}(K)$. It is an element of $\operatorname{B}(H\otimes \overline{H})\bar{\otimes} \operatorname{B}(K\otimes \overline{K})$, hence it acts on the Hilbert space $\operatorname{HS}(H\otimes K)$ via $T(R)=\sum_{n=1}^{N} (A_n\otimes \mathbb{1}_{K}) R (\mathbb{1}_{H}\otimes B_n)$.
I have the following question: does the map $TJTJ\colon \operatorname{HS}(H\otimes K)\rightarrow \operatorname{HS}(H\otimes K)$ preserve the cone of positive operators? I.e. if $R$ is a positive Hilbert Schmidt operator, is $\sum_{n,m=1}^{N} (A_n\otimes B_m^*) R (A_m^* \otimes B_n)$ also positive?
This question is related to the determination of the cone (in the sense of standard form of von Neumann algebra) related to the weight $\varphi\otimes\psi'$, where $\varphi$ is the trace on $\operatorname{B}(H)$ and $\psi'$ is the trace on $(\operatorname{B}(K)\otimes\mathbb{1}_{\overline{K}})'=\mathbb{1}_{K}\otimes\operatorname{B}(\overline{K})$ (commutant calculated in $\operatorname{B}(\operatorname{HS}(K))$), so I would also be interested in any information related to this problem.