Moduli spaces of one dimensional sheaves on log surfaces and Hilbert schemes

Let $X$ be a smooth projective rational surface, $D\subset X$ an effective anticanonical curve, $\beta$ a curve class on $X$ and $\mathfrak{d}=\sum w_iP_i$ an effective divisor on $D_{\mathrm{sm}}$. We consider the moduli space $\mathcal{M}_\beta(X, D, \mathfrak{d})$ of sheaves on $X$ which are direct images of rank-$1$ torsion-free sheaves on integral curves $C$ in $\beta$ such that $C|_D=\mathfrak{d}$, and show that each point of $\mathcal{M}_\beta(X, D, \mathfrak{d})$ is smooth over a point in the product of the Hilbert schemes of surface singularities of types $A_{w_i-1}$. Hence, $\mathcal{M}_\beta(X, D, \mathfrak{d})$ has symplectic singularities and admits a unique symplectic resolution.