We state that a locally $(n-1)$-connected compactum with integral cohomological dimension $n$ has $n$-cohomological dimension modulo $p$ for some prime $p$. As a consequence, the integral cohomological dimension of the square of such a space is $2n$. In particular, the dimension of the square of an $n$-dimensional, locally $(n-1)$-connected compactum is $2n$.
"Dimension of the square of a compactum and local connectedness." Proc. Japan Acad. Ser. A Math. Sci. 78 (6) 69 - 71, June 2002. https://doi.org/10.3792/pjaa.78.69