Edge-label ratio conjecture for triangular grids

Let $T_1(k,c)$ be the triangular grid, with $k$ and $c$ as in the tail asymptotics conjecture. For indices satisfying the displayed ranges, define the edge-label ratios

$$r_{2,1}(c,r,d)=\frac{\langle r,d,2\rangle_c}{\langle r,d,1\rangle_c},\qquad r_{3,1}(c,r,d)=\frac{\langle r,d,3\rangle_c}{\langle r,d,1\rangle_c},$$

and

$$x(c,r)=\frac{\langle r,1,1\rangle_c}{\langle r-1,1,1\rangle_c},\qquad y(r,d)=\frac{\langle r,d,1\rangle_c}{\langle r,d-1,1\rangle_c}.$$

Edge-label ratio conjecture. For the $T_1(k,c)$ grid,

$$r_{2,1}(c,r,d)\asymp\frac{2(r-d)+1}{2d-1},\qquad 1\le d\le r\le c,$$ $$r_{3,1}(c,r,d)\asymp\frac{2(c-r)+1}{2d-1},\qquad 1\le d\le r\le c,$$ $$x(c,r)\asymp\frac{r-1}{2r-1}\frac{2(c-r)+3}{(c-r)+1},\qquad 2\le r\le c,$$ $$y(r,d)\asymp\frac{d-1}{2d-3}\frac{2(r-d)+3}{(r-d)+1},\qquad 2\le d\le r\le c.$$

The conjecture aims to completely describe limiting ratios of certain edge labels as $k$ grows while $c$ remains constant. It is motivated by viewing the labels as products of factors with numerators and denominators that are linear functions of at most three variables; the supplied text gives no resolution.