Compressed primitivity problem in free groups

For a fixed integer $r\ge 2$, we prove that the \emph{compressed primitivity problem} in the free group $F_r=F(x_1,\dots,x_r)$ is decidable in non-deterministic polynomial time. That is, for a \emph{straight-line program} $\mathcal A$ over $\{x_1,\dots,x_r\}^{\pm1}$ representing an element $g\in F_r