Extension of the mixed double-dimer construction to all self-intersecting contours

Let

$$\{z_{i,j,k}^{(4)}: \left(-k < i,j, i+j < k-1 \mathrm{~for~} k > 0\right) \mathrm{~~and~~} \left( k-1 < i,j, i+j < -k \mathrm{~for~} k \leq 0 \right) \}$$

be the three-dimensional space of toric cluster variables described in the source. For self-intersecting contours, consider subgraphs with vertices of multiplicity one and two, using mixed configurations in which vertices have multiplicity $1$ except for a simply-connected interior region of vertices of multiplicity $2$. Extension conjecture. The construction rule inspired by the family $\varphi\left(A(0,0,n+2)\right)=z_{0,0,n+1}^{(4)}$ extends to all other self-intersecting contours, with an analogue of the taut condition for crossing the inner double-dimer region, and degenerates to the usual dimer-only rule when the self-intersection disappears, namely when $i$, $j$ or $i+j$ falls outside of $[-k,k-1]$ or $[k-1,k]$. The source gives no evidence of resolution.