The class-number conjecture for pseudo-Anosov maps with fixed dilatation

Let $(\Lambda,[I],K)$ be the triple corresponding to a pseudo-Anosov map $\phi\in Mod~(X)$, and let $\lambda_\phi$ be its dilatation. The integral order $\Lambda$ has class number $h_\Lambda=|\Lambda/[I]|$. Class-number conjecture. The number of conjugacy classes of pseudo-Anosov automorphisms with dilatation $\lambda_\phi$ is equal to $h_\Lambda$. This proposes an operator-algebraic arithmetic invariant for distinguishing pseudo-Anosov conjugacy classes with the same dilatation; the source gives no resolution of the claim.