Generalized extinction scenario for positive symmetric matrices

Let ${\bf A}$ be a positive symmetric matrix, and let the model satisfy the hypotheses and notation of Theorem~, including the condition

modified suitably for ${\bf A}$. The negative-definite condition

and the existence of an interior equilibrium condition

$$are not assumed. **Generalized extinction conjecture.** With a suitable modification of condition$$

, the conclusions of Theorem~ should hold for a general positive symmetric matrix ${\bf A}$.

The theorem describes metastable extinction in the infinite-population limit, with the first boundary hit typically removing exactly one component belonging to the fittest set and with the expected extinction time diverging. The authors specifically expect that the negative-definite and interior-equilibrium assumptions can be removed in the general positive symmetric setting; no proof is supplied.