Ross-symbol form of the weak Beilinson conjecture for hypergeometric elliptic curves

Let $\alpha\in{\mathbb Q}\setminus\{0,1\}$ and let $X_\alpha$ be the elliptic curve in the preceding construction. Let $\xi_{\mathrm{Ross}}\in K_2(X_\alpha)^{(2)}$ be the higher Ross symbol, assumed to be integral, and let $\mathscr F_{\frac12,\frac12}(\alpha)$ denote the hypergeometric function used in the source. Ross-symbol Beilinson conjecture. If $\xi_{\mathrm{Ross}}$ is integral, then

$$\operatorname{Re}[\mathscr F_{\frac12,\frac12}(\alpha)]\sim_{{\mathbb Q}^\times}L'(X_\alpha,0).$$

This is obtained in the paper by evaluating the regulator of the Ross symbol and combining that evaluation with the weak Beilinson prediction; it remains conditional on the integrality assumption and the conjectural regulator formula.