Let $NC(n)$ denote the lattice of noncrossing partitions of $n$, and let $G$ denote the Hasse diagram of $NC(n)$ with respect to covering relations, viewed as an undirected graph.

I'm interested in the connectivity of graphs of geodesics between vertices of $G$. If $p$ and $q$ are two noncrossing partitions of $n$, let $G_{p,q}$ be the graph whose vertices are minimal-length paths (i.e. geodesics) in $G$ from $p$ to $q$, and where an edge is drawn between two paths $\alpha$ and $\beta$ when $\alpha$ and $\beta$ differ in precisely one element.

Is $G_{p,q}$ always connected, or is there an example of $(n,p,q)$ such that it's not?

If, for example, $p = id$ and $q = (12\ldots n)$ (as noncrossing permutations), then geodesics from $p$ to $q$ are maximal chains from $p$ to $q$, and in this case $G_{p,q}$ is known to be connected (e.g. Bessis "The dual braid monoid" Prop. 1.6.1, http://arxiv.org/pdf/math/0101158.pdf).

But how about if $p$ and $q$ are not comparable in the partial order? Is anything known about the connectivity of $G_{p,q}$?

EDIT: A definition and basic discussion of $NC(n)$ can be found at http://en.wikipedia.org/wiki/Noncrossing_partition.

EDIT 2: Using a computer program in Mathematica (with help from the package posets.m by Curtis Greene et al, http://www.haverford.edu/math/cgreene/posets.html), I've checked that $G_{p,q}$ is always connected for $n < 7$.

set$[n]=\{1,2,\dots,n\}$? $\endgroup$ – bof Dec 22 '14 at 19:23