One-unknown word equations and three-unknown constant-free word equations

We prove connections between one-unknown word equations and three-unknown constant-free word equations, and use them to prove that the number of equations in an independent system of three-unknown constant-free equations is at most logarithmic with respect to the length of the shortest equation in the system. We also study two well-known conjectures. The first conjecture claims that there is a constant c such that every one-unknown equation has either infinitely many solutions or at most c. The second conjecture claims that there is a constant c such that every independent system of three-unknown constant-free equations with a nonperiodic solution is of size at most c. We prove that the first conjecture implies the second one, possibly for a different constant.