I will present an explicit finite-dimensional model for the derived moduli stack of flat connections on $\mathbb{C}^{k}$ with logarithmic singularities along a weighted homogeneous Saito free divisor. I will focus in particular on the example of plane curve singularities of the form $x^p = y^q$. These moduli spaces are conjectured to admit shifted Poisson structures. I will discuss this conjectural picture and present some partial results. This talk will be based on the preprints arXiv:2209.00631, arXiv:2301.00962.