The Magnus trace map surjectivity conjecture for SL(m,C) character varieties

Let $\mathfrak{X}_r$ be the $\mathrm{SL}(m,\mathbb{C})$ character variety of a free group of rank $r$, and let $\mathcal{I}\subset\{\operatorname{tr}(\mathbf{X}_{i_1}\mathbf{X}_{i_2}\cdots\mathbf{X}_{i_l})\}$ be a set of parameters for $\mathfrak{X}_r$. The associated Magnus trace map is

$$\mathsf{MagTr}_{\mathcal{I}}:\mathfrak{X}_r\to\mathbb{C}^{|\mathcal{I}|}.$$

Magnus trace map surjectivity conjecture. The map $\mathsf{MagTr}_{\mathcal{I}}$ is surjective only for the cases

$$(m,r)=(1,r),\quad (m,1),\quad (2,2),\quad (2,3),\quad (3,2).$$

The conjecture asserts that, apart from these cases, the parameter space is not fully realized by trace coordinates. The preceding discussion notes that the map is surjective in the listed low-rank cases, while Florentino showed non-surjectivity for $\mathrm{SL}(2,\mathbb{C})$ when $r\geq4$; the general assertion for $m\geq3$ and $r\geq2$, excluding $(3,2)$, remains open.