The heat-flow conjecture for average lozenge orientations in dimer systems

Let $\bar\Omega$ be the limiting region with red boundary segments corresponding to constrained boundary conditions and blue segments corresponding to free boundary conditions. Let $O_1^{(n)},\dotsc,O_k^{(n)}$ be translated finite unions of unit triangles in the interior of $\Omega_n$, shrinking to distinct points $a_1,\dotsc,a_k\in\Omega$. Let $\boldsymbol{F}$ be the field of average lozenge orientations, and let $\boldsymbol{h}$ be the steady-state heat-flow vector field for the corresponding system with point sources at the $a_i$ and the stated insulating and constant-temperature boundary conditions. The heat-flow conjecture. There is a scalar constant $c$, depending on $\Omega$, the portions of the boundary with each boundary condition, and the shapes of the $O_i^{(n)}$, but not on the common temperature or the limiting points $a_1,\dotsc,a_k$, such that

$$\lim_{n\to\infty} n\\,\boldsymbol{F}\sim c\\,\boldsymbol{h}.$$

This conjectures that the macroscopic average orientation field of the dimer system is proportional to the heat-flow field of the associated steady-state physical system. The source provides no resolution, so the status is left open.