Generic existence and uniqueness of means in tree shape spaces

Let $\bar{X}$, $\bar{\bar{X}}$, $\bar{Z}$, and $\bar{\bar{Z}}$ be the tree shape spaces introduced in the paper, and let $x_1,\ldots,x_n$ be points in one of these spaces. A set of points is generic when it avoids the nongeneric configurations where the local $CAT(0)$ property fails.

Generic means conjecture. For a generic set of points $x_1,\ldots,x_n$ in $\bar{X}$, $\bar{\bar{X}}$, $\bar{Z}$, or $\bar{\bar{Z}}$, means exist and are unique.

The conjecture proposes that the existence and uniqueness results established at points within $CAT(0)$ injectivity neighborhoods extend to generic configurations, despite the failure of the $CAT(0)$ property at certain highly nongeneric points. The supplied text does not state whether this conjecture has been resolved.