Strong reciprocity conjecture for the infinitesimal Chow dilogarithm

Assume that $k$ is an algebraically closed field, let $C/k$ be a smooth projective curve, and let

$$\Gamma(K,3): B_{3}(K)\to B_{2}(K)\otimes K^{\times}\to\Lambda^{3}K^{\times}$$

be the Bloch complex, with residue maps summed over all closed points giving

$$res_{|C|}:\Gamma(k(C),3)\to\Gamma(k,2).$$

Here $h$ is required to vanish on $k^{\times}\wedge\Lambda^{2}k(C)^{\times}$.

Strong reciprocity conjecture. The residue map $res_{|C|}:\Gamma(k(C),3)\to\Gamma(k,2)$ is homotopic to zero. More precisely, there exists a map $h$ fitting into a commutative diagram whose restriction to $k^{\times}\wedge\Lambda^{2}k(C)^{\times}$ is zero:

\xymatrix{B_3(k(C)) \ar[r] & B_2(k(C))\otimes k(C)^{\times} \ar[r] ^-{ \Delta}\ar[d] ^{res_{|C|}}&\Lambda^{3}k(C)^{\times} \ar[d]^{res_{|C|}} \ar@{.>}[dl] _{h} \\ & B_{2}(k) \ar[r]^{\delta} & \Lambda^{2}k^{\times}.}

This strengthens the consequence of Suslin reciprocity that the image of the residue map in $\Lambda^{2}k^{\times}$ lies in the image of $\delta$, and would provide homotopy-level reciprocity for the weight-three Bloch complex. The statement is presented as a conjecture in the source, and its resolution is not established by the supplied text.