Goncharov's spanning conjecture for motivic iterated integrals over number fields

Let $K\subset\mathbb{C}$ be a number field. The ring $\mathcal{H}(K)$ is the ring of effective motivic periods of the category of mixed Tate motives over $K$, and motivic iterated integrals on $\mathbb{P}^{1}\setminus\{\infty\}\cup K$ are the motivic lifts of iterated integrals whose marked points lie in $K$. Goncharov's conjecture. $\mathcal{H}(K)$ is spanned by motivic iterated integrals on $\mathbb{P}^{1}\setminus\{\infty\}\cup K$. This conjecture predicts that all effective motivic periods over a number field are generated, as a rational vector space, by motivic iterated integrals with singularities in that field. It is open for any number field $K$.