Beilinson's $L$-value conjecture for singular K3 surfaces with Picard rank 20

Let $X$ be a singular $K3$ surface with Picard rank $20$ over $\mathbb{Q}$, let $H^3_\mathcal{M}(X,\mathbb{Q}(4))_{\operatorname{tr}}$ be the transcendental motivic cohomology, and let $H_2(X(\mathbb{C}),\mathbb{Q})_{\operatorname{tr}}^-$ be the anti-invariant transcendental homology. Let $f$ be the weight-three newform associated with $X$ by the modularity theorem stated in the source. Beilinson's conjecture. For every nontrivial $\xi\in H^3_\mathcal{M}(X,\mathbb{Q}(4))_{\operatorname{tr}}$ and a generator $\gamma$ of $H_2(X(\mathbb{C}),\mathbb{Q})_{\operatorname{tr}}^-$,

$$\frac{1}{(2\pi i)^3}\left<\gamma,\operatorname{reg}_{X,\operatorname{tr}}(\xi)\right>_{X,\operatorname{tr}}=a\,L'(f,-1),\qquad a\in\mathbb{Q}^\times.$$

This is the modular-form formulation of the preceding regulator prediction, using the equality between the transcendental motive's $L$-function and the newform's $L$-function. The source gives no resolution of this conjecture.