Total non-hyperbolicity conjecture for minimal unimodal combinatorics

Let a minimal unimodal combinatorics be a minimal combinatorial type in the unimodal region, and call it unobstructed when it is realized by the corresponding quadratic dynamical system without a Thurston obstruction. Total non-hyperbolicity conjecture. Every minimal unimodal combinatorics is unobstructed, even in the totally non-hyperbolic case. The claim would follow from the preceding conjecture because reaching the ideal boundary within the unimodal region would require passing through the shift locus; the source also suggests that a suitable modification of the bone argument might prove it, but gives no proof.