On Gubeladze's nilpotence conjecture

Let $M$ be a reasonable commutative monoid written multiplicatively and $R$ a commutative ring. Then the natural numbers act on the monoid algebra $R[M]$, with $n$ acting by raising elements of $M$ to the $n-th$ power. Consider a functor $E$ from rings to spectra; then the natural numbers act on $E(R[M])/E(R)$ by the induced maps, which we call dilations. Gubeladze conjectured that if $R$ is regular and $E = K$ is algebraic $K-theory$, then this action is nilpotent. He proved his conjecture whenever $R$ contains the rational numbers. A different proof of this result was later given by the speaker jointly with Cortinas, Walker and Weibel and they adapted their proof to all regular $R $ containing any field. I will discuss this proof and outline how the approach can be used to prove the general case of the conjecture.