Does entanglement of a bipartite PPT state $rho$ imply entanglement of $rho + rho^{Gamma}$?

Consider an entangled bipartite quantum state $rho in mathcal{M}_d(mathbb{C}) otimes mathcal{M}_{d’}(mathbb{C})$ which is positive under partial transposition, i.e., $rho^Gamma geq 0$. As separability of $rho$ is equivalent to separability of its partial transpose $rho^Gamma$, we know that $rho^Gamma$ is entangled. Does this imply that the sum $rho + rho^Gamma$ (ignoring trace normalization) is also entangled? If not, can we impose restrictions on $rho$ which guarantee that the above proposition holds?

In the language of entanglement witnesses, the problem reduces to finding a common witness that detects both $rho$ and $rho^Gamma$. Let $W$ be the entanglement witness detecting $rho$, i.e., $text{Tr} (Wrho) < 0$. Then $W$ is non-decomposable (as $rho$ is PPT) and is of the canonical form $P+Q^Gamma – epsilon mathbb{I}$, where $P, Q geq 0$ are such that $text{range}(P) subseteqtext{ker}(delta)$ and $text{range}(Q) subseteq text{ker}(delta^Gamma)$ for some bipartite edge state $delta$ (these are special states that violate the range criterion for separability in an extreme manner, see edge states) and $0 < epsilon leq text{inf}_{|e,frangle} langle e,f | P+Q^Gamma | e,f rangle$. If $delta$ is such that $text{ker}(delta) cap text{ker}(delta^Gamma)$ is not empty, then we can choose $P=Q$ to be the orthogonal projector on $text{ker}(delta) cap text{ker}(delta^Gamma)$, in which case $W=W^Gamma$ is the common witness. But is this always true? Can we use optimization of entanglement witness to ensure this condition?

---

Cross posted on math.SE

Cross posted on quantumcomputing.SE