Beilinson's conjecture for motivic cohomology of genus-one curves

Let $C$ be a smooth projective curve over $\mathbb Q$ of genus $1$, let $E$ be its Jacobian, and let $\alpha$ be a nontrivial element of $H^2_\mathcal M(C,\mathbb Q(3))$. Let $\gamma_C^+$ be a generator of $H_1(C(\mathbb C),\mathbb Q)^+$. Beilinson's conjecture. There is an $a\in\mathbb Q^\times$ such that

$$\frac{1}{(2\pi i)^2}\int_{\gamma_C^+}\operatorname{reg}_C(\alpha)=aL'(E,-1).$$

This is the genus-one case of Beilinson's conjecture, relating regulators of motivic cohomology classes to special values of $L$-functions. It is presented as a conjectural input for the paper's Mahler-measure identities.