Recall that the Kronecker product $s_\lambda * s_\mu$ of two Schur functions $s_\lambda$ and $s_\mu$ is the symmetric function whose expansion (in terms of Schur functions) is given by
\begin{equation} \sum_{\nu \, \vdash \, n} g_{\lambda \mu}^\nu \, s_\nu \end{equation}
where $\lambda$, $\mu$, and $\nu$ are partitions of $n$ and $g_{\lambda \mu}^\nu$ is the Kronecker coefficient, which famously counts the multiplicity of $V_\nu$ in the tensor product $V_\lambda \otimes V_\mu$ of the irreducible representations of the symmetric group $S_n$.
Switch now to the quasi-symmetric world: Given a composition $\alpha = (\alpha_1, \dots, \alpha_k)$ of $n$ let $L_\alpha$ be the fundamental quasi-symmetric functions defined by
\begin{equation} L_\alpha = \sum x_{\ell_1} \cdots \,x_{\ell_k} \end{equation}
where the sum is taken over all sequences $1 \leq \ell_1 \leq \cdots \leq \ell_k$ such that $\ell_i < \ell_{i+1}$ whenever $i = \alpha_1 + \cdots + \alpha_j$ for some $1 \leq j \leq k-1$.
The space of symmetric functions within the $\mathrm{QSym}_n :=\Bbb{Q}$-span of $\{ L_\alpha \, \big| \, \alpha \, \models \, n \}$ coincides with the $\mathrm{Sym}_n:= \Bbb{Q}$-span of $\{ s_\lambda \, \big| \, \lambda \, \vdash \, n \}$ the latter of which is endowed with the Kronecker $*$-product.
Question: Can the Kronecker $*$-product on $\mathrm{Sym}_n$ be extended to all of $\mathrm{QSym}_n$ so that there exist non-negative integers $\tilde{g}_{\alpha,\beta}^{\, \gamma}$ for each triple $\alpha$, $\beta$, $\gamma$ of compositions of $n$ satisfying
\begin{equation} L_\alpha * L_\beta = \sum_{\gamma \, \models \, n} \tilde{g}_{\alpha,\beta}^{\, \gamma} \, L_\gamma \quad \text{?} \end{equation}
p.s. Covertly, I am asking whether or not there is some kind of tensor product structure (as in a symmetric tensor category) on the projective indecomposable representations of the 0-Hecke algebra $H_n(0)$. Any thoughts on that would also be appreciated.
thanks, ines.