Existence of detectably non-Hausdorff manifolds without bifurcate curves

Let $M$ be a manifold. A bifurcate curve in $M$ is a pair of smooth injective curves that agree up to a parameter value and then disagree at that value and thereafter. A manifold is detectably non-Hausdorff when its algebra of smooth functions is not isomorphic to that of any Hausdorff manifold. Existence conjecture. There exist detectably non-Hausdorff manifolds with no bifurcate curves. This would clarify whether the absence of bifurcate curves can coexist with detectable non-Hausdorffness, a condition the paper discusses as potentially relevant to the physical reasonableness of non-Hausdorff spacetimes. The source does not report a resolution.