Proposition: Second-iterate failure need not persist forever

Assume a PPT pair $F,G:M_d(\mathbb{C})\to M_d(\mathbb{C})$ has entangled $J(GF)$. There is a trace-preserving PPT channel $\Xi :M_{3d+1}(\mathbb{C})\to M_{3d+1}(\mathbb{C})$ such that

Proof. Replace the two-block cycle in (eq:completion) by the acyclic transitions

Send the missing trace from each transition to the absorbing flag, and discard input off-diagonal blocks. The same branchwise proof gives a PPT channel. The input-block-$0$, output-block-$2$ corner of its square is $stJ(GF)$. After three steps every input block has reached the flag. Trace preservation identifies the remaining scalar with $\mathrm{Tr}X$.