On dimensionally exotic maps

We call a value $y=f(x)$ of a map $f:X\to Y$ dimensionally regular if $\dim X\le \dim(Y\times f^{-1}(y))$. 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 $\dim X\ge 6$ admits a map $f:X\to Y$ without dimensionally regular values. Also we exhibit a 4-dimensional Boltyanskii compactum for which every map has a dimensionally regular value.