Vanishing Ones Conjecture for transformed triangular grids

Let $T_{n,k}$ denote the triangular grid obtained after $k$ transformations from the $n$-grid, and let $T_{n,n-s}^s$ denote an $s$-subgrid of $T_{n,n-s}$. For an edge, its edge value is the corresponding resistance label, and $Interior(T_{n,n-s}^s)$ denotes the interior of the $s$-subgrid. For integer $n\geq 1$, the following holds. Vanishing Ones Conjecture.

(a) $T_{n,n}^1=1$.

(b) For $1\leq s\leq \left\lfloor\frac{n+1}{4}\right\rfloor$, $Interior(T_{n,n-s}^s)$ is equal to one.

(c) With $s$ as in (b), for an edge in the complement of the interior of an $s$-subgrid, its edge value is strictly less than one if it lies on the edge boundary of some $s'$-grid, $1\leq s'<s$; its edge value is strictly greater than one otherwise.

(d) For any $s$ with $\left\lfloor\frac{n+1}{4}\right\rfloor<s\leq n-1$, there are no edges with label $1$ in $T_{n,n-s}$; the ones “vanish”.

The conjecture formalizes the observed pattern of edge resistances under repeated transformations of regular triangular grids, including the disappearance of edges with resistance one after sufficiently many reductions.