Finite-bound conjecture for independent three-variable word-equation systems

Let a system of equations be a set of word equations, and call it independent if it is not equivalent to any proper subset. A system is constant-free when its equations contain only variables, and a solution is nonperiodic when it is not periodic. Consider independent systems of constant-free equations in three variables having a nonperiodic solution. Finite-bound conjecture. There exists a number $c$ such that every such system has size at most $c$. This is explicitly presented as a weaker conjecture than the claim that the largest examples have size two. The paper proves a bound of $17$, while the conjectured bound of two remains open; consequently, the existence of some finite bound is open in the source's stated context.