Common-factor conjecture for numerators of triangular-grid edge resistances

Let $L_s^{(i)}$, for $i\in\{1,2\}$, be the edge-resistance quantities defined in the paper, and let $\operatorname{num}(x)$ denote the numerator of the maximally reduced fractional representation of $x$. Let $\gcd$ denote the greatest common divisor. Common-factor conjecture. For every integer $s\geq 1$,

$$\gcd\bigl(\operatorname{num}(L_s^{(1)}),\operatorname{num}(L_{s+1}^{(2)})\bigr)>1.$$

This conjecture predicts a persistent arithmetic relation between numerators of resistance values in successive transformed rims. The supplied text gives no evidence that it has been proved or disproved.