Positive Lyapunov exponents for quotient maps outside the sink basin

Let $X$ be the vector field from Theorem 3.2, let $\mathcal{U}$ be its neighborhood in $\mathfrak{X}^2(M)$, and for $Y\in\mathcal{U}$ let $g_Y$ be the quotient map. Let $p(Y)$ be the periodic sink and let its stable manifold be understood in the quotient dynamics. Positive-exponent quotient-map conjecture. If $g_Y$ does not send the critical set inside the stable manifold of $p(Y)$, then the complement of this stable manifold contains the basin of a physical measure for $g_Y$ whose Lyapunov exponents are positive. This predicts a physical expanding component whenever the critical set avoids the sink basin; the source does not report a proof or resolution.