Volume-preserving singular adapted metric conjecture

Let $X$ be a conservative $C^1$ vector field and let $\Gamma$ be a singular hyperbolic set for $X$. A singular adapted metric is a Riemannian metric satisfying the contraction, domination, and sectional expansion conditions associated with the singular hyperbolic splitting. Volume-preserving singular adapted metric conjecture. There exists a singular adapted metric whose induced volume form is preserved by the flow of $X$. The preceding results establish existence of a singular adapted metric, but do not show that its induced volume form is preserved; this volume-preserving refinement therefore remains open.