Let $P\to \Sigma$ be a $\mathrm{SU}(2)$-principal bundle over a smooth oriented closed genus $g\geq 2$ real Riemannian surface $\Sigma$. Let $\mathcal{A}$ be the space of connexions over $P$. Let $\Omega^k:=\Omega^k(\Sigma;\mathrm{Ad}P)$ be the space of $\mathrm{Ad}P$-valued differential $k$-forms on $\Sigma$ (here $\mathrm{Ad}P=P\times_\mathrm{Ad}\mathfrak{su}(2)$ is the adjoint bundle). For $A\in \mathcal{A}$, let $\mathrm{d}_A:\Omega^k\to\Omega^{k+1}$ be the covariant exterior derivative, $\delta_A:\Omega^k\to \Omega^{k-1}$ be the adjoint operator of $\mathrm{d}_A$ and $\Delta_A:=\mathrm{d}_A\delta_A+\delta_A\mathrm{d}_A$ be the Laplacian.
Question : If $A$ is irreducible, what is the dimension of the space of harmonic $\mathrm{Ad}P$-valued differential 1-forms ? i.e. : $$ \dim \ker(\Delta_A|_{\Omega^1}) \quad = \quad \text{?} $$
If $A$ is flat, the dimension is $6g-6$. I'm looking for the non-flat scenario. I think I read somewhere that the non-flat case is also $6g-6$, but I don't remember where I read that and why this is such. All I'm sure of is that $\ker(\Delta_A|_{\Omega^1})$ is even dimensional. Also, I already asked for $\dim \ker(\Delta_A|_{\Omega^1})$ as an added side question at the very end of this lengthy unanswered previous post. Since this was an edit added two weeks after the main question I don't think this is a duplicate but could be a self-contained post.