Nontriviality of the BV cycle in the Pachner complex

Let $\mathsf{c}_\Delta^k$ be the BV cycle defined by the displayed construction referred to as (c_Delta^n). BV-cycle nontriviality conjecture. The cycle $\mathsf{c}_\Delta^k$ is nontrivial. The source gives no resolution of this claim; in the surrounding discussion it also notes that the vanishing of its square is conjectured in the appropriate full Pachner complex.