Tight extension conjecture for valuations

Let $R$ be the local domain, $nu$ a valuation centered on $R$, $cal T$ the relevant tree of local blowings up, and $tilde H'_0$ the corresponding implicit prime ideal in the completion $hat R'$. Write $cal P_\beta$ for the $nu$-ideal of value $\beta$ and $\operatorname{gr}_\nu R'$ for the associated graded algebra.

Tight extension conjecture. The valuation $nu$ admits at least one tight extension $hat\nu_-$. It can be chosen so that, for rings $R'$ sufficiently far in $cal T$,

$$\hat\nu_-\left(\frac{\hat R'}{\tilde H'_0}\setminus\{0\}\right)=\nu(R'\setminus\{0\}),$$

and, for every $\beta\in\nu(R'\setminus\{0\})$, its $\hat\nu_-$-ideal of value $\beta$ is

$$\frac{\mathcal P_\beta\hat R'}{\tilde H'_0}.$$

In particular,

$$\operatorname{gr}_\nu R'=\operatorname{gr}_{\hat\nu_-}\frac{\hat R'}{\tilde H'_0}.$$

The conjecture strengthens an earlier conjecture in the paper and is intended to support extension of valuations to completions and applications to local uniformization. The supplied text says that local uniformization in full generality remains open, but gives no direct resolution of this conjecture.