Quantitative intrinsic connectivity conjecture for semilinear positive phases

Let $f\in C_0^\infty(\mathbb{R})$ satisfy $f\ge0$ and $f(s)=0$ for all $s\le0$. Let $u$ solve

$$\Delta u=f(u)\quad\text{in }\mathbb{R}^3,$$

and suppose that the blow-down limit

$$U(x)=\lim_{\epsilon\to0^+}\epsilon\,u(x/\epsilon)$$

exists and has connected positive phase $\{x\in\mathbb{R}^3:U(x)>0\}$. Quantitative intrinsic connectivity conjecture. The minimum path distance between points in $\{x\in\mathbb{R}^3:u(x)>0\}$ is comparable to their Euclidean straight-line distance, with a comparison constant depending only on $U$, or perhaps even an absolute constant.

The conjecture asserts that connectedness of the blow-down positive phase prevents arbitrarily thin or long intrinsic bottlenecks in the positive phase of $u$. The source presents the universal-constant formulation as a possible strengthening, and gives no evidence of resolution.