Conjecture on elliptic-unit generators of ray class fields

Let $K$ be an imaginary quadratic field, let $\mathfrak f$ be an ideal of $\mathcal O_K$, and let $K_{\mathfrak f}$ be the corresponding ray class field. Under the conditions of the main theorem, there are $u_{\mathfrak c},v_{\mathfrak c},u_1,v_1\in\frac{1}{f}\mathbf Z$ and an elliptic unit

$$\epsilon(\mathfrak c)=\frac{\phi(u_{\mathfrak c},v_{\mathfrak c},\tau_1)}{\phi(u_1,v_1,\tau_1)}\in K_{\mathfrak f},$$

where $f$ is the smallest positive integer in $\mathfrak f$. Elliptic-unit generator conjecture. The elliptic unit $\epsilon(\mathfrak c)$ generates $K_{\mathfrak f}$ over $K$ if the conditions of the main theorem hold. The authors report that all their experiments support the claim and that the proof of a cited lemma suggests it, but the supplied text does not state that it has been proved.