  Assign independent weights to the edges of the square lattice, from the
uniform distribution on $\{a,b\}$ for some $0<a<b<\infty$. The weighted graph
induces a random metric on $\mathbb{Z}^2$. Let $T_n$ denote the distance
between $(0,0)$ and $(n,0)$ in this metric. The distribution of $T_n$ has a
well-defined median. Itai Benjamini asked in 2011 if the sequence of Boolean
functions encoding whether $T_n$ exceeds its median is noise sensitive? In this
paper we present the first progress on Benjamini's problem. More precisely, we
study the minimal weight along any path crossing an $n\times n$-square
horizontally and whose vertical fluctuation is smaller than $n^{1/22}$, and
show that for this observable, 'being above the median' is a noise sensitive
property.
