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

Let $A$ be the polynomial algebra in $r$ variables with coefficients in an algebraically closed field $k$. When the characteristic of $k$ is $2$, Carlsson conjectured that any $\mathrm{dg}$-$A$-module that is free of rank $N$ as an $A$-module and whose homology is nontrivial and finite dimensional as a $k$-vector space satisfies $N\geq 2^r$. In this paper, we examine a stronger conjecture concerning varieties of square-zero upper triangular $N\times N$ matrices. Stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when $N = 8$ without any restriction on the characteristic of $k$. This result also verifies that if $X$ is a product of $3$ spheres of any dimensions, then the elementary abelian $2$-group of rank $4$ cannot act freely on $X$.