  Kruskal's theorem states that a sum of product tensors constitutes a unique
tensor rank decomposition if the so-called k-ranks of the product tensors are
large. In this work, we propose a conjecture in which the k-rank condition of
Kruskal's theorem is weakened to the standard notion of rank, and the
conclusion is relaxed to a statement on the linear dependence of the product
tensors. Our conjecture would imply a generalization of Kruskal's theorem.
Several adaptations and generalizations of Kruskal's theorem have already been
obtained, but these results still cannot certify uniqueness when the k-ranks
are below a certain threshold. Our generalization would contain several of
these results, and could certify uniqueness below this threshold. We prove our
conjecture over an arbitrary field $\mathbb{F}$ when the underlying
multipartite vector space takes any one of three forms:
${\mathbb{F}^{d_1}\otimes \mathbb{F}^{d_2}},
\;{\mathbb{F}^{d_1}\otimes\mathbb{F}^{d_2}\otimes \mathbb{F}^2,}$ or
$\mathbb{F}^{d_1}\otimes \mathbb{F}^2 \otimes\cdots \otimes \mathbb{F}^2$. As a
corollary to the third case, we prove that if $n$ product tensors form a
circuit, then they have rank greater than one in at most $n-2$ subsystems. This
is a quadratic improvement over a recent bound obtained by Ballico, and is
sharp.
