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It was shown in \\cite{first-exotic} that if a map $f:X\\to Y$ between compact metric spaces does not have dimensionally regular values, then $X$ is a Boltyanskii compactum, i.e. a compactum satisfying the equality $\\dim(X\\times X)=2\\dim X-1$. In this paper we prove that every Boltyanskii compactum $X$ of dimension $\\dim X \\geq 6$ admits a map $f:X\\to Y$ without dimensionally regular values. 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