The primitive-word and measure-preservation conjecture for free groups

Let $\mathbf{F}_{k}$ be the free group of rank $k$. A nontrivial word $w\in\mathbf{F}_{k}$ is primitive if it belongs to some free basis. A finitely generated subgroup $H\le\mathbf{F}_{k}$ is measure preserving if, for every finite group $G$, a uniformly random homomorphism $\alpha_{G}\colon\mathbf{F}_{k}\to G$ restricts to a uniformly distributed element of $\operatorname{Hom}(H,G)$. Write $H\stackrel{*}{\le}\mathbf{F}_{k}$ when $H$ is a free-factor subgroup. The primitive-word and measure-preservation conjecture. For every $w\in\mathbf{F}_{k}$,

$$w\textrm{ is primitive }\Longleftrightarrow w\textrm{ is measure preserving}.$$

More generally, for $H\le_{\mathrm{fg}}\mathbf{F}_{k}$,

$$H\stackrel{*}{\le}\mathbf{F}_{k}\Longleftrightarrow H\textrm{ is measure preserving}.$$

Primitivity and free factorness imply measure preservation; the conjecture asserts the converse for words and finitely generated subgroups. The source attributes this converse to several authors, but no resolution is supplied here.