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Context. Understanding planetesimal formation is an essential first step towards understanding planet formation. The distribution of these first solid bodies drives the locations where planetary embryos, which eventually form fully-fledged planets, grow. Aims. We seek to understand the parameter space of possible protoplanetary disk formation and evolution models of our Solar System. A good protoplanetary disk scenario for the Solar System must meet at least the following three criteria: (1) It must produce an extended gas and dust disk (e.g. 45 au for the dust); (2) within the disk, the local dust-to-gas ratio in at least two distinct locations must sufficiently increase to explain the early formation of the parent bodies of non-carbonaceous and carbonaceous iron meteorites; and (3) dust particles, which have condensed at high temperatures (i.e. calcium–aluminium-rich inclusions), must be transported to the outer disk. Though current protoplanetary disk models are able to satisfy one or two of these criteria, none have been successful in recreating all three. We aim to find scenarios that satisfy all three. Methods. In this study we used a 1D disk model that tracks the evolution of the gas and dust disks. Planetesimals are formed within the disk at locations where the streaming instability can be triggered. We explored a large parameter space to study the effect of the disk viscosity, the timescale of infall of material into the disk, the distance within which material is deposited into the disk, and the fragmentation threshold of dust particles. Results. We find that scenarios with a large initial disk viscosity (α > 0.05), a relatively short infall timescale (Tinfall < 100–200kyr), and a small centrifugal radius (RC~ 0.4 au; i.e. the distance within which material falls into the disk) result in disks that satisfy all three criteria needed to represent the protoplanetary disk of the Solar System. The large initial viscosity and short infall timescale result in a rapid initial expansion of the disk, which we dub the 'inflationary phase' of the disk. Furthermore, a temperature-dependent fragmentation threshold, which accounts for cold icy particles breaking more easily, results in larger and more massive disks. This, in turn, results in more 'icy' than 'rocky' planetesimals. Such scenarios are also better in line with our Solar System, which has small terrestrial planets and massive giant planet cores. Finally, we find that scenarios with large RC cannot transport calcium–aluminium-rich inclusions to the outer disk and do not produce planetesimals at two locations within the disk. [ABSTRACT FROM AUTHOR]