Density conjecture for $C^1$-conjugacy classes in $\hat D_{\alpha,\beta}$

Let $\alpha>1$ and $0<\beta<1$, and let $\hat D_{\alpha,\beta}\subset\operatorname{Diff}^1([0,1])$ be the set of diffeomorphisms whose fixed points are precisely $0$ and $1$, with $Df(0)=\alpha$ and $Df(1)=\beta$. Density conjecture. Every $C^1$-conjugacy class is dense in $\hat D_{\alpha,\beta}$. Equivalently, for any $f,g\in\hat D_{\alpha,\beta}$, there is a diffeomorphism arbitrarily $C^1$-close to $g$ that is conjugated to $f$ by a diffeomorphism of $[0,1]$. The conjecture concerns the density of smooth conjugacy classes in this fixed-derivative space; the source provides no resolution.