  We present the positive-partial-transpose squared conjecture introduced by M.
Christandl at Banff International Research Station Workshop: Operator
Structures in Quantum Information Theory (Banff International Research Station,
Alberta, 2012). We investigate the conjecture in higher dimensions and offer
two novel approaches (decomposition and composition of quantum channels) and
correspondingly, several schemes for finding counterexamples to this
conjecture. One of the schemes involving the composition of PPT quantum
channels in unsolved dimensions yields a potential counterexample.
