  In this paper we further explore the recently discovered connection by
Bj\"{o}rklund and Kaski [STOC 2024] and Pratt [STOC 2024] between the
asymptotic rank conjecture of Strassen [Progr. Math. 1994] and the three-way
partitioning problem. We show that under the asymptotic rank conjecture, the
chromatic number of an $n$-vertex graph can be computed deterministically in
$O(1.99982^n)$ time, thus giving a conditional answer to a question of Zamir
[ICALP 2021], and questioning the optimality of the $2^n\operatorname{poly}(n)$
time algorithm for chromatic number by Bj\"{o}rklund, Husfeldt, and Koivisto
[SICOMP 2009].
  Our technique is a combination of earlier algorithms for detecting
$k$-colorings for small $k$ and enumerating $k$-colorable subgraphs, with an
extension and derandomisation of Pratt's tensor-based algorithm for balanced
three-way partitioning to the unbalanced case.
