Strong Whitney-reversible property

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Let $P$ be a topological property. Then $P$ is said to be a strong Whitney-reversible property provided whenever $X$ is a continuum such that $\mu^{-1}(t)$ has property $P$ for some Whitney map $\mu$ of $C(X)$ and all $0 < t < \mu(X)$, then $X$ has property $P$.