The twisted Teichmüller curve uniqueness conjecture

Let $K$ be the real quadratic field associated with the fundamental discriminant $D$, and let $d4aa_D$ be its ring of integers. A matrix $M$ has relative prime entries when its entries generate the unit ideal. A twisted Teichmüller curve is the curve defined by such a matrix, and its determinant is $d4\det(M)$.

Twisted Teichmüller curve uniqueness conjecture. Suppose $D\equiv 5\pmod 8$ is a fundamental discriminant with narrow class number $h_D^+=1$. For every $n\ind4aa_D$, all matrices

$$M\in\operatorname{GL}_2^+(K)\cap\operatorname{Mat}^{2\times 2}(d4aa_D)$$

of determinant $n$ with relative prime entries define the same twisted Teichmüller curve; equivalently, there is exactly one twisted Teichmüller curve of determinant $n$.

The claim would extend the proved uniqueness result for prime determinants to arbitrary determinants. The paper reports strong numerical evidence, including determinants whose prime divisors have all possible splitting behaviors, but does not prove the assertion in general.