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Let $A$ be the polynomial algebra in $r$ variables with coefficients in analgebraically closed field $k$. When the characteristic of $k$ is $2$, Carlssonconjectured that for any $\mathrm{dg}$-$A$-module $M$, which has dimension $N$as a free $A$-module, if the homology of $M$ is nontrivial and finitedimensional as a $k$-vector space, then $N\geq 2^r$. Here we examine a strongerconjecture concerning varieties of square-zero upper triangular $N\times N$matrices with entries in $A$. Stratifying these varieties via Borel orbits, weshow that the stronger conjecture holds when $N = 8$ without any restriction onthe characteristic of $k$. This result also verifies that if $X$ is a productof $3$ spheres of any dimensions, then the elementary abelian $2$-group oforder $4$ cannot act freely on $X$.

The rank 8 case of a conjecture on square-zero upper triangular matrices