Conjecture on expressible elements in cyclic groups

Let $A$ be a cyclic group of order $m$. For $\boldsymbol{\pi}=(\pi_1,\ldots,\pi_m)\in(\mathbf{Z}/m\mathbf{Z})^m$ with $\pi_1\ge\cdots\ge\pi_m$, an element $x\in A$ is $\boldsymbol{\pi}$-expressible if, for some ordering $g_1,\ldots,g_m$ of the elements of $A$, one has

$$x=g_1^{\pi_1}\cdots g_m^{\pi_m}.$$

Let $\boldsymbol{\pi}'$ be obtained by subtracting $\pi_m$ from every part of $\boldsymbol{\pi}$, so that its last part is $0$. Cyclic-group expressibility conjecture. Every element of $A$ is $\boldsymbol{\pi}$-expressible unless either

1. $\boldsymbol{\pi}'=(m-r,r,0,\ldots,0)$ for some $r$, or
2. there exists an integer $p>1$ dividing every part of $\boldsymbol{\pi}'$ and also dividing $m$.

This conjecture gives a proposed classification of when all elements of a cyclic group occur as products determined by an exponent vector; the paper presents it as being supported by numerical evidence, while the stated exceptional cases are not asserted to be exhaustive in the source beyond this conjectural claim.