Positive-coefficient conjecture for adjacent edges in independence polynomials

Let $\mathsf K_n$ be the complete graph and let $Z(\mathbf I\mathsf K_n;\mathbf y)$ be the partition function of its independent-set system. For edges $e,f$, let $\Delta Z\{e,f\}$ denote the Rayleigh difference. Adjacent-edge coefficient conjecture. If $e$ and $f$ are adjacent edges in $\mathsf K_n$, then $\Delta Z(\mathbf I\mathsf K_n)\{e,f\}$ has nonnegative coefficients. The claim has been checked for $n\leq6$ and is open in general.