Universal analytic Gröbner basis conjecture

Let $K\{\mathbf{X};P\}$ be the analytic power-series ring over the valued field $K$, with $P$ the parameter set of valuation vectors, and let $I\subset K\{\mathbf{X};P\}$ be an ideal. A finite set $G\subset I$ is a universal analytic Gröbner basis (UAGB) if it is an $\mathbf{r}$-local Gröbner basis of $I$ for every $\mathbf{r}\in P$. Universal analytic Gröbner basis conjecture. Every ideal $I\subset K\{\mathbf{X};P\}$ has a UAGB; equivalently, there is a finite set $G\subset I$ such that $G$ is an $\mathbf{r}$-local Gröbner basis of $I$ for every $\mathbf{r}\in P$. The preceding proposition proves finiteness of the relevant leading terms for principal ideals, while the conjecture asks for the analogous finite-basis statement for arbitrary ideals and is presented as the stronger end goal.