Generic nonexistence of analytic geodesics in analytic function spaces

Let $\mathcal{H}$ be the space of Kähler metrics on $M$. Let $E$ and $E'$ be analytic function spaces on $M$, meaning Banach spaces of analytic functions containing all analytic functions with sufficiently small radius of injectivity.

Nonexistence conjecture for analytic geodesics. There exists an open dense subset of $(E\cap\mathcal{H})^2$ such that no pair of elements in this subset is linked by a geodesic in $E'$.

This conjecture predicts generic failure of analytic geodesic connectivity and is presented as opposite in direction to Chen's conjecture for smooth metrics. It would imply, after choosing a countable sequence of suitable analytic function spaces $E'$, a residual set of pairs not linked by an analytic geodesic. Its status is not resolved in the supplied text.