The profinite primitivity conjecture for free groups

Let $\mathbf{F}_{k}$ be the free group of rank $k$, and let $\widehat{\mathbf{F}}_{k}$ be its free profinite completion. An element is primitive if it belongs to a basis of the relevant free group. The profinite primitivity conjecture. For every word $w\in\mathbf{F}_{k}$,

$$w\text{ is primitive in }\widehat{\mathbf{F}}_{k}\Longleftrightarrow w\text{ is primitive in }\mathbf{F}_{k}.$$

The source notes that primitivity in the discrete free group implies primitivity in its free profinite completion, and conjectures the converse. No resolution is supplied here.