Somekawa's generalized Galois symbol injectivity conjecture

Let $A_1,\ldots,A_r$ be abelian varieties over a perfect field $K$, and let $n$ be an integer invertible in $K$. The generalized Galois symbol is

$$s_n:K(K;A_1,\ldots,A_r)/n\longrightarrow H^r(K,A_1[n]\otimes\cdots\otimes A_r[n]).$$

Somekawa's conjecture. The generalized Galois symbol $s_n$ is always injective.

This conjecture concerns the relationship between Somekawa $K$-groups and Galois cohomology; its status is not determined by the supplied text.