Dust substructures in protoplanetary discs can be signatures of embedded young planets whose detection and characterisation would provide a better understanding of planet formation. Traditional techniques used to link substructures' morphology to the properties of putative embedded planets present several limitations that the use of deep learning methods has partly overcome. In our previous work, we developed DBNets, a tool exploiting an ensemble of Convolutional Neural Networks (CNNs) to estimate the mass of putative planets in disc dust substructures. This inference problem, however, is degenerate as planets of different masses could produce the same rings and gaps if other physical disc properties were different. In this paper, we address this issue improving our simulation-based inference pipeline to estimate the full posterior distribution for the planet mass and three additional disc properties: the disc $\alpha$-viscosity, the scale height and the dust Stokes number. We also address some minor issues of our previous tool. The new pipeline involves a CNN that summarises the input images in a set of summary statistics, followed by an ensemble of normalising flows that model the inferred posterior for the target properties. We tested our pipeline on a dedicated set of synthetic observations using the TARP test and standard metrics, demonstrating its accuracy and precision. Additionally, we use the results obtained on the test set to study the degeneracies between pairs of parameters. Finally, we apply the developed pipeline to a set of 49 gaps in 34 protoplanetary discs' continuum observations. The results show typically low values of $\alpha$-viscosity, disc scale heights, and planet masses, with 83% of them being lower than 1M$_J$. These low masses are consistent with the non-detections of these putative planets in direct imaging surveys. Our tool is publicly available.