General convergence conjecture for diffusion processes in narrow tubes

Let $\rho_j$ denote the radius scale of the ball associated with vertex $O_j$, and partition the vertices into the small-ball, intermediate-ball, and large-ball cases, denoted by $\mathfrak{S}$, $\mathfrak{M}$, and $\mathfrak{L}$, respectively. Let $\Pi^\epsilon \circ Z^\epsilon$ be the projected diffusion process in the narrow tube, and let $\bar{Z}$ be the continuous Markov process on the metric graph $\Gamma$ whose generator $\bar{L}$ acts as the second derivative on the edges and satisfies the gluing conditions

$$\sum_{k:I_k \sim O_j} \frac{\lambda_k^{d-1}}{\sum_{l:I_l \sim O_j} \lambda_l^{d-1} } \frac{d f}{d x_k}(O_j) =0, \quad j \in \mathfrak{S},$$ $$\sum_{k:I_k \sim O_j} \frac{\lambda_k^{d-1} }{\sum_{l:I_l \sim O_j} \lambda_l^{d-1} } \frac{d f}{d x_k}(O_j) = \frac{\rho_j^{d} V_d}{\sum_{l:I_l \sim O_j} \lambda_l^{d-1} V_{d-1}} \frac{d^2 f}{d x^2}(O_j), \quad j \in \mathfrak{M},$$ $$\frac{d^2 f}{d x^2}(O_j) =0, \quad j \in \mathfrak{L}.$$

General convergence conjecture. The process $\Pi^\epsilon \circ Z^\epsilon$ converges weakly to $\bar{Z}$ in $C([0,\infty);\Gamma)$, uniformly with respect to the initial condition.

This conjecture extends the known weak convergence result for the regime $r_j(\epsilon) \asymp \epsilon$, in which the limiting process has weighted Kirchhoff gluing conditions and no vertex waiting. It predicts distinct limiting graph diffusions for small, intermediate, and large balls, with the corresponding gluing conditions above.