Nikolaev's rank conjecture for CM elliptic curves and RM noncommutative tori

Let ${\cal E}_{CM}^{(-D,f)}$ be an elliptic curve with complex multiplication by the order of conductor $f$ in the imaginary quadratic field, and let ${\cal A}_{RM}^{(D,f)}$ be the corresponding noncommutative torus with real multiplication. Write $rk({\cal E}_{CM}^{(-D,f)})$ for the rank of the elliptic curve and $c({\cal A}_{RM}^{(D,f)})$ for the arithmetic complexity, namely the number of independent variables in the minimal period of the continued fraction of $\sqrt D$. The Teichmüller functor relates these objects by $F({\cal E}_{CM}^{(-D,f)})={\cal A}_{RM}^{(D,f)}$. Nikolaev's rank conjecture.

$$rk({\cal E}_{CM}^{(-D,f)})+1=c({\cal A}_{RM}^{(D,f)}).$$

The conjecture proposes a correspondence between the Mordell–Weil rank of a CM elliptic curve and the arithmetic complexity of its associated RM noncommutative torus; the supplied text gives no resolution evidence.