Say that a compact $K\subset\mathbb R^d$ has the Wada property if there are disjoint, connected open sets $V_1,V_2,\ldots,V_n\subset\mathbb R^d$, $n\geq3$, such that $K=\partial V_1=\partial V_2=\ldots=\partial V_n$. In conversation with someone, they mentioned something along the lines of: in the plane, no point of such a $K$ (or perhaps it was 'not many points' in the sense of Hausdorff dimension) could be accessible from all of the $V_i$ simultaneously, but for $d>2$, this need not be the case. I haven't been able to find any sources on Wada domains coming close to this claim (most papers I have seen seem to be preoccupied with $d=2$, and with physical applications rather than the geometry of $K$). Does anyone have a good source for this, or more generally for the theory of accessible points in Wada domain boundaries?