Monotonicity and positivity conjecture for Welschinger invariants on toric Del Pezzo surfaces

Let $\Sigma$ be a toric Del Pezzo surface equipped with its tautological real structure, and let $D$ be an ample divisor on $\Sigma$. For admissible values of $m$, let $W_{D,m}$ denote the corresponding Welschinger invariant.

Positivity and monotonicity conjecture. The Welschinger invariants $W_{D,m}$ are positive if $m<[(c_1(\Sigma)\cdot D-1)/2]$, non-negative if $m=[(c_1(\Sigma)\cdot D-1)/2]$, and satisfy

$$W_{D,m-1}\ge W_{D,m}\quad\text{for}\quad m\ge 1.$$

This conjecture predicts both positivity up to the maximal relevant number of pairs of imaginary points and monotonicity as that number increases. The supplied text does not state whether the conjecture has been resolved.