Conjecture on weak omega-limit sets of Volterra quadratic stochastic operators

Let $V\in\tilde{\mathcal V}^-$ and let $\frak{f}(V)=(a_{ki})$. For ${\mathbf{x}}\in S$, write $\operatorname{supp}({\mathbf{x}})$ for its support, let ${\mathbf e}_j$ denote the $j$-th unit vector, and let ${\mathbf 0}$ denote the zero vector. Assume that $a_{ki}<0$ for all $k<i$. Weak omega-limit set conjecture. For every ${\mathbf{x}}\in S$:

1. if $|\operatorname{supp}({\mathbf{x}})|<\infty$, then

$$\omega_V^{(w)}({\mathbf{x}})=\left\{{\mathbf e}_{\max\{\operatorname{supp}({\mathbf{x}})\}}\right\};$$

2. if $|\operatorname{supp}({\mathbf{x}})|=\infty$, then

$$\omega_V^{(w)}({\mathbf{x}})=\left\{{\mathbf 0}\right\}.$$

This conjecture predicts the weak asymptotic behavior of Volterra quadratic stochastic operators with strictly negative coefficients above the diagonal: finite-support points converge weakly to the unit vector indexed by their largest support index, whereas infinite-support points converge weakly to the zero vector. The input supplies no evidence resolving the conjecture, so its status remains open.