The small shuffle algebra action conjecture for the entire K-theory

Let $S$ be the surface under consideration, let $\CV_\emph{sm}$ be its small shuffle algebra, and let $K_\CM$ denote the entire K-theory group in place of the tautological part $K_\CM'$. Under Assumption A, the action is understood as an abelian group homomorphism assigning to each $R(z_1,\ldots,z_k)\in\CV_\emph{sm}$ a homomorphism from $K_\CM$ to $K_{\CM\times S^k}$, with composition governed by the shuffle product as in Corollary. Small shuffle algebra action conjecture. Under Assumption A, there is an action

$$\CV_\emph{sm}\curvearrowright K_\CM$$

where

$$\delta\left(\frac{z_1}{z}\right)$$

acts as $e(z)$ of the defining formula for $e(z)$. The notion of action is defined as in Corollary. This is the expected extension of the action from the tautological K-theory group under Assumption B to the entire K-theory group under Assumption A; the supplied text gives no resolution, so the conjecture remains open.