The Merino–Welsh inequality for composition enumerators

Let $\bar\gamma(\underline{a};x,y)$ be the composition-indexed quantity used in the source. Let $(a_0,a_,\ldots,a_r)$ be a composition with $a_0=0$ and $a_r\geq 2$. Merino–Welsh conjecture for composition enumerators.

$$\bar\gamma(\underline{a};2,0)+\bar\gamma(\underline{a};0,2)\geq 2\bar\gamma(\underline{a};1,1),$$

with equality if and only if $\underline{a}=(0,2,2,\ldots,2)$. The source proposes this as a stronger conjecture intended to imply the matroidal Merino–Welsh conjecture; the notation and status are not otherwise established in the supplied excerpt.