Universal polynomial and positivity conjecture for wall-crossing coefficients

Let $S$ be a rational surface, let $N,d,r$ be the parameters in the wall-crossing formula, let $K^2$ denote $K_S^2$, and let $Q_i(N,d,r,K^2)$ and $P_i(N,d,r,K^2)$ be the polynomials appearing there. Universal polynomial and positivity conjecture. For every $i$, one has

$$Q_i(N,d,r,K^2)=P_i(N,d,r,K^2)+R_i(N,d,r,K^2),$$

where $R_i(N,d,r,K^2)$ is a polynomial in $N,d,r,K^2$ of degree $i-2$. When $R_i$ is regarded as a polynomial in $N,-d,-r,K^2$, all its coefficients are positive; the same is true of $Q_i$. This conjecture is a stronger form of the wall-crossing-shape conjecture mentioned immediately beforehand and is supported by the computations in the paper. Its general validity is not resolved in the supplied text.