Positivity conjecture for Welschinger invariants of $(\mathbb{R}P^1)^3$

Let $(\mathbb{R}P^1)^3$ be the product of three real projective lines with the standard real structure, and let $\mathfrak{s}_{(\mathbb{R}P^1)^3}$ be the $Spin_3$-structure over $(\mathbb{R}P^1)^3$ specified in Theorem 2-W. For $(a,b,c)\in\mathbb{N}^*\times\mathbb{N}\times\mathbb{N}$ and $l\in\left\{0,\ldots,\frac{1}{2}(a+b+c-1)\right\}$, write $W_{(\mathbb{R}P^1)^3}^{\mathfrak{s}_{(\mathbb{R}P^1)^3},\mathfrak{o}_{(\mathbb{R}P^1)^3}}((a,b,c),l)$ for the corresponding genus-$0$ Welschinger invariant. Positivity conjecture. For all such triplets $(a,b,c)$ and all such $l$, one has

$$W_{(\mathbb{R}P^1)^3}^{\mathfrak{s}_{(\mathbb{R}P^1)^3},\mathfrak{o}_{(\mathbb{R}P^1)^3}}((a,b,c),l)\geq 0.$$

The conjecture is motivated by the computations in the paper's table, which exhibit non-vanishing invariants in the cases not ruled out by the preceding vanishing proposition. Its general validity for all indicated triplets and values of $l$ is not established in the supplied text.