Positivity conjecture for normalized characters in twisted Boolean cumulants

For positive integers $k_1,\ldots,k_l$, let $\Sigma_{k_1,\ldots,k_l}$ denote the normalized character and let $\hat{B}_i$ denote the twisted Boolean cumulants. Positivity conjecture for twisted Boolean cumulants. The normalized character can be expressed as a polynomial in the twisted Boolean cumulants $\hat{B}_i$ with coefficients that are non-negative integers. This conjecture extends the observed positivity of the displayed examples and asks whether the positivity phenomenon persists when the shifted parameter is $\zeta=0$. Its resolution is not specified in the source.