The irreducibility characterization of primitive nonnegative tensors

Let $\mathbb{A}$ be a nonnegative tensor of order $m$ and dimension $n$. A tensor is irreducible if there is no nonempty proper subset $I\subset\{1,\ldots,n\}$ such that

$$c_{i_1\ldots i_m}=0\qquad\forall i_1\in I,\ \forall i_2,\ldots,i_m\notin I.$$

A tensor is $j$-primitive when its $j$-primitive degree is finite, equivalently when some tensor power has the relevant positivity in column $j$.

Irreducibility characterization conjecture. A nonnegative tensor $\mathbb{A}$ is primitive if and only if $\mathbb{A}$ is irreducible and there exists some $j\in[n]$ such that $\mathbb{A}$ is $j$-primitive.

For matrices, the corresponding characterization follows from strong connectivity of the associated digraph and the period-one condition. The source proposes the tensor analogue; no resolution is given in the supplied text.