Nikolaev's conjecture relating arithmetic complexity to the rank of a CM elliptic curve

Let $E_{CM}$ be an elliptic curve with complex multiplication, let ${\theta}_{E_{CM}}$ be the real quadratic irrational associated with $E_{CM}$, and let ${\bb A}_{{\theta}_{E_{CM}}}$ be the corresponding noncommutative torus with real multiplication. Denote by $c({\bb A}_{{\theta}_{E_{CM}}})$ the length of the minimal period of the regular continued fraction of ${\theta}_{E_{CM}}$, and by $rk~(E_{CM})$ the rank of $E_{CM}$. Nikolaev's rank conjecture. One has

$$c({\bb A}_{{\theta}_{E_{CM}}})=rk~(E_{CM})+1.$$

This conjecture relates the arithmetic complexity of the noncommutative torus associated with a complex-multiplication elliptic curve to the Mordell–Weil rank of the curve. The supplied text gives no evidence of resolution, so its status remains open.