The gamma-cap conjecture for Poisson fields of two variables

Let $K=K\{{\mathbf{f}}\}$ be a Poisson field, let ${^1\Gamma}_0(K)$ be the $1$-valuation gamma-cap, and let $\operatorname{fht}(K)$ denote the flag height. Gamma-cap conjecture. If

$$\operatorname{fht}(K)\geq 4,$$

then

$${^1\Gamma}_0(K)\neq\Bbbk.$$

This predicts that sufficiently large flag height forces the $1$-valuation gamma-cap to contain elements beyond the base field. The source supplies no resolution, so the conjecture remains open.