Steprāns–Watson conjecture on Baumgartner axioms in finite powers of the reals

Let $cmathsf{BA}$ denote Baumgartner's axiom for the real line, and let $cmathsf{BA}(X)$ be its parametrized version for a topological space $X$. For a positive integer $n$, write $cmathbb R^n$ for the $n$-dimensional Euclidean space. Steprāns–Watson conjecture. $cmathsf{BA}$ implies $cmathsf{BA}(cmathbb R^n)$ for every finite $n>1$. The conjecture asks whether the Baumgartner axiom for the real line implies its higher-dimensional analogues; the converse implication is known not to follow from the higher-dimensional axioms, while this direction remains open.