I am reading the paper [1], where the author proves that the skein module of links in a handlebody $F\times I$ has a free basis given by products $D_1 \cdots D_n$ where each $D_i$ is the closure of $v_i\in \pi_1(F)$, $v_i$ is the smallest length lexicographically smallest representative of its conjugacy class and $v_1\leq v_2\leq\cdots\leq v_n$. Besides several instances where the author makes a vague argument which presumably can be fixed using diamond lemma, I am stuck in the point where he proves that such products span the skein module.

The only place devoted to this issue is the second paragraph of "Proof of Theorem 2.11", p. 333. He says that, as an algebra both the skein module $\mathcal{S}(F\times I)$ and the symmetric algebra over the span of conjugacy classes of the fundamental group $S(R\hat{\pi}^\circ)$ are generated by the representatives of conjugacy classes of $\pi_1(F)$, hence the claim. But the map between these is not an algebra homomorphism, so this argument does not apply.

I don't know how to fix this gap. Naively one would think that induction on the number of crossings would work. But here is the problem. We need to be able to do two things: 1) switch the order of components, 2) change a representative of a component (i.e. conjugate the element $v_i\in\pi_1(F)$ to make it lexicographically smaller). Now the problem with these two operations is that they behave well with respect to different invariants. 1) works up to diagrams with smaller number of crossings, 2) works up to diagrams whose components have words of smaller lengths (see Lemma 1.7). Clearly 1) messes up the maximal length, indeed, usually the left-overs will have words whose lengths are sums of lengths of what we start with. On the other hand, 2) involves performing some isotopies. Even if the number of self-crossing of the component does not go up under such isotopies, the number of crossings with other components may increase. So we are chasing our own tail here.

UPDATE: First note that the question is about modules over the polynomial ring $\mathbb{Z}[v,v^{-1},h]$, not over the field of fractions. It turns out the question is not that easy. I did some experiments and it turns out the statement is actually false in almost all cases, except as stated by the author. For instance, if the surface is not planar, i.e. for the punctured torus it is false. If it is planar, i.e. a disk with several disks removed, and if the order of generators is different, or the choice of generators is different, it is false. The order of generators plays a role in saying what's lexicographically smaller. For instance, take a disk with $2$ disks removed. Suppose the generators of the fundamental group are $x$ going around one puncture, and $y$ going around both punctures. If the order on the generators is $x<y<y^{-1}<x^{-1}$, then the statement is false. It still seems to be true if the order of generators is as the author describes, namely $x<x^{-1}<y<y^{-1}$. Why?

**Some explanations.** We choose a base point $*$ on the boundary of $F$. For any element $\gamma\in \pi_1(F,*)$ we represent it by a path $\gamma:[0,1]\rightarrow F$ such that $\gamma(0)=\gamma(1)=*$ but $\gamma(t)\neq *$ for $0<t<1$. The graph of $\gamma$ is a path in $F\times [0,1]$ connecting $(*,0)$ to $(*,1)$. We close it up by adding the line $*\times [0,1]$. The resulting map from $S^1$ to $F\times [0,1]$ is a knot. More explicitly, if you view $S^1$ as $[0,1]/(0\sim 1)$ then $s:S^1\to F\times[0,1]$ is defined by
$$
s(t)=\begin{cases}
(\gamma(2 t), 2t) & (t\leq \frac12),\\
(*, 2-2t) & (t\geq \frac12).
\end{cases}
$$
This is what I called *closure* of $\gamma\in \pi_1(F)$. For several elements $v_1,\ldots,v_n\in \pi_1(F)$ we stack the corresponding knot diagrams on top of each other and obtain what I denoted by $D_1\cdots D_n$.

Speaking about elements of $\pi_1(F,*)$ suppose we represent $\pi_1(F)$ as a free group on generators $g_1,\ldots,g_k$. Suppose we choose an order on the set $g_1,\ldots,g_k,g_1^{-1},\ldots,g_k^{-1}$. Then we define an order on $\pi_1(F)$ as follows. If we want to compare $v_1,v_2\in\pi_1(F)$ we write $v_1,v_2$ as a reduced word in letters $g_i$ and $g_i^{-1}$ and say that $v_1<v_2$ if the length of $v_1$ is strictly less than that of $v_2$ or if they have same length and the word of $v_1$ is lexicographically smaller than the word of $v_2$.

UPDATE: There is an interesting idea of inducting on the length of the knot with respect to a metric of negative curvature. Here is an example. Let $K$ be any knot in $F\times I$ represented by a geodesic. Suppose $K$ has several crossings. Take $K^n=K\cdot K \cdot \cdots\cdot K$, $n$ times. Choose an arc of $K$. Then over this arc there is $n$ arcs of $K^n$. Let us replace these $n$ arcs by some braid which lifts a cyclic permutation. We obtain a knot, denote $K_n'$, its diagram is again formed by geodesics in the limit. Now how do we commute $K_n'$ past $K_m'$?

[1] Przytycki, J. (1992). Skein module of links in a handlebody. Topology '90 (pp. 315-342).