In the $d$-dimensional hypercube bin packing problem, a given list of $d$-dimensional hypercubes must be packed into the smallest number of hypercube bins. Epstein and van Stee [SIAM J. Comput. 35 (2005)] showed that the asymptotic performance ratio $ρ$ of the online bounded space variant is $Ω(\log d)$ and $O(d/\log d)$, and conjectured that it is $Θ(\log d)$. We show that $ρ$ is in fact $Θ(d/\log d)$, using probabilistic arguments.