Nikolaev's Q-rank conjecture for CM elliptic curves

Let $p$ be a prime with $p=3\pmod 4$, let ${\cal E}(p)={\cal E}_{CM}^{(-p,1)}$ be the associated $\mathbb Q$-curve, and let $h_K$ be the class number of the relevant imaginary quadratic field. Let ${\cal A}_{RM}^{(p,1)}$ be the corresponding noncommutative torus with real multiplication, and let $c({\cal A}_{RM}^{(p,1)})$ denote its arithmetic complexity. The $\mathbb Q$-rank of ${\cal E}(p)$ is $rk_{\mathbb Q}({\cal E}(p))=\frac{1}{2h_K}rk({\cal E}(p))$. Nikolaev's $\mathbb Q$-rank conjecture.

$$\frac{1}{2h_K}rk({\cal E}(p))+1=c({\cal A}_{RM}^{(p,1)}).$$

This is presented as a refinement of the rank conjecture for the additional symmetry of $\mathbb Q$-curves; the supplied text gives no resolution evidence.