Sahi's positivity conjecture for coefficients of the geometric-mean expansion

Let $L$ be an FKG poset, and let $f_1(x),f_2(x),\ldots$ be a sequence of positive monotone decreasing functions on $L$. Form the formal power series

$$F(x,t)=1-f_1(x)t-f_2(x)t^2-\cdots$$

and define its geometric mean by

$$G(F)=\exp\bigl({\mathord{\cal E}}(\log F)\bigr)=1-c_1t-c_2t^2-\cdots.$$

Sahi's coefficient-positivity conjecture. For every $n\geq 1$, the coefficient $c_n$ satisfies

$$c_n\geq 0.$$

The conjecture is a formal-power-series reformulation of the generalized inequalities $E_n\geq0$, motivated by potential applications of the FKG inequality in probability, combinatorics, statistics, and physics. Its resolution is not given in the source.