Diagonal-positivity criterion for three-variable symmetric rational functions

Let $e_1,e_2,e_3$ be the elementary symmetric polynomials in three variables, let

$$Q=1-e_1+a e_2+b e_3,$$

and set $F=1/Q$. Let $\operatorname{diag}_F$ denote the diagonal power series of $F$, and let $a_0\approx-1.81$ be characterized by

$$6(1-a_0)=2-3a_0+2(1-a_0)^{3/2}.$$

Diagonal-positivity conjecture. The series $\operatorname{diag}_F$ is nonnegative if and only if

$$b\leq\begin{cases}6(1-a)&a\leq a_0,\\2-3a+2(1-a)^{3/2}&a_0\leq a\leq1,\\-a^3&a\geq1. \end{cases}$$

This is presented as an explicit prediction derived from conjectures about positivity in three variables; the source provides supporting results in lower dimensions and additional evidence, but the full three-variable assertion remains open.