Gubeladze's nilpotence conjecture for monoid algebras

Let $k$ be a commutative ring, let $M$ be an abelian monoid, and let $\mathfrak c=(c_1,c_2,\ldots)$ be a sequence of integers with $c_i\ge 2$. For each $c\ge 2$, the dilation $\theta_c:k[M]\to k[M]$ is defined by $\theta_c(\chi_a)=\chi_{a^c}$, and write

$$K_*(k[M])^{\mathfrak c}=\operatorname*{colim}_{\theta_{c_n}}K_*(k[M]).$$

Assume that $k$ is regular Noetherian and that $M$ is cancellative, torsion-free, and has no non-trivial units. Gubeladze's nilpotence conjecture.

$$K_*(k[M])^{\mathfrak c}=K_*(k).$$

This predicts that all nilpotent contributions disappear after iterated dilations; the supplied text gives no resolution status.