Conjecture on long-time concentration in a dominant eigenspace of the interaction matrix

Let $B$ and $V$ be weight matrices satisfying Assumption~, with $V\neq\mathrm{Id}$, and let $(\rho_t^\beta)_{t\geq 0}$ be the token-distribution evolution on ${\mathbb S}^{d-1}$. Let $F$ be the eigenspace of $V$ associated with its largest eigenvalue, and let $F^{\mathrm{abs}}$ be the eigenspace associated with an eigenvalue of largest magnitude. For a probability measure $\rho_\infty\in\mathcal{P}({\mathbb S}^{d-1})$, write $\operatorname{supp}(\rho_\infty)$ for its support. Dominant-eigenspace concentration conjecture. There exist weight matrices $B$ and $V\neq\mathrm{Id}$ satisfying Assumption~, and a probability measure $\rho_\infty\in\mathcal{P}({\mathbb S}^{d-1})$ supported on $F\cap{\mathbb S}^{d-1}$ or $F^{\mathrm{abs}}\cap{\mathbb S}^{d-1}$ such that

$$\lim_{t\to\infty}W_2(\rho_t^\beta,\rho_\infty)=0.$$

Numerical evidence and asymptotic stability results support convergence toward a dominant eigenspace at long times, beyond the $\mathcal{O}(\log\beta)$ time scale. The conjecture is trivial when $V=\mathrm{Id}$, while the nontrivial case concerns general interactions with $V\neq\mathrm{Id}$; the supplied text does not establish the claim.