Validity conjecture for reciprocal-distance weight functions on augmented hypercubes

For a positive integer $n\geq 3$, let $Q_{n-1}$ be the $(n-1)$-dimensional hypercube, and let $G_n$ be the graph obtained by attaching a root $r$ to an arbitrary vertex of $Q_{n-1}$ through an extra pendant edge. Let $d(v,r)$ denote graph distance from $v$ to $r$, and define a weight function $w_n$ on $G_n$ by

$$w_n(r)=0,\qquad w_n(v)=\frac{1}{d(v,r)}\quad\text{for all }v\neq r.$$

Validity conjecture. The weight function $w_n$ is valid.

This claim concerns the construction of non-tree weight functions used to obtain pebbling bounds for hypercubes. The supplied text does not state whether the claim has been proved or remains open.