Classification conjecture for Baumgartner axioms on perfect Polish spaces

Let $X$ be a perfect Polish space, meaning a separable completely metrizable space with no isolated points. Let $cmathsf{BA}(X)$ denote the Baumgartner axiom parametrized by $X$, and let $cmathfrak p$ be the pseudointersection number. A closed nowhere dense set $F\table X$ is invariant under all self-homeomorphisms of $X$ when every homeomorphism $h:X\to X$ satisfies $h[F]=F$. Classification conjecture for Baumgartner axioms. Exactly one of the following holds: (1) $X$ contains a closed nowhere dense $F\table X$ invariant under every self-homeomorphism of $X$, and consequently $cmathsf{BA}(X)$ provably fails; (2) this obstruction fails and $X$ is not one-dimensional, in which case $cmathsf{BA}(X)$ is equivalent to $cmathfrak p>cmathaleph_1$; or (3) this obstruction fails and $X$ is one-dimensional, in which case $cmathsf{BA}(X)$ is equivalent to $cmathsf{BA}$. Moreover, if $X$ is as in (3) and $Y$ is as in (2), then $cmathsf{BA}(X)$ implies $cmathsf{BA}(Y)$. The conjecture proposes a classification of the full Baumgartner axiom for perfect Polish spaces. The invariant closed nowhere dense set is a topological obstruction forcing failure, and the claim is that this obstruction characterizes provable failure; the remaining cases are distinguished by dimension and by their relationship with $cmathfrak p>cmathaleph_1$ and the real-line axiom.