## 2 Materials and Methods
### Rate Estimates  
In order to evidence the differences among competing processes following the collisions, sufficient statistics have to be collected. The TTF involving lighter fragment mass \(A>23\), for the \({}^{40}\)Ar + \({}^{208}\)Pb reaction at 230 MeV, has been evaluated to be around 5 mb. This value is determined by considering a TTF to BF cross section ratio of about 0.7%, according to the \({}^{40}\)Ar-induced reactions systematics with a beam energy of 230 MeV on \({}^{208}\)Pb target [49; 51], and a fusion-fission cross section of 700 mb, calculated with the Bass model. Furthermore, we considered the sharing among the possible tripartitions energetically allowed. Although this is a very large number, by considering the increase in the probability of emitting two doubly-magic nuclei per fission decay and \(A_{3}>23\), we can expect a branching of 1% for the \({}^{132}\)Sn + \({}^{68}\)Zn + \({}^{48}\)Ca tripartition. This tripartition, indeed, coincides with one of the deepest minima of TFP built by considering both DTF and STF mechanisms, as discussed in Section 2.2. So we can estimate a cross section of 50 \(\upmu\)b for this specific tripartition. This gross estimate is corroborated by the 9.4 \(\upmu\)b of \({}^{28}\)Mg in coincidence with two fission fragments measured in the \({}^{4}\)He + \({}^{238}\)U reaction at \(E^{*}=118\) MeV [74]. This value, according to the systematics [23], is, indeed, expected to be lower than ours because it involves tripartitions with only one doubly-magic nucleus, and it is performed at lower beam energy and in a composite system with lower fissility parameter (Z\({}^{2}\)/A), 36.6 instead of 40.3 for the \({}^{40}\)Ar + \({}^{208}\)Pb reaction.  
Along with the production yield, the detection efficiency also has to be considered. The proposed setup is intended for the collection of both double and triple coincidence events. The double events are necessary for the BF yield and can be used to provide a description of the TTF by estimating the properties of missing fragments, as done in several previous experiments. However, a definitive identification of TTF requires the direct and simultaneous measurement of three fragments. By considering the experimental efficiency of 4% for double coincidence events, typical of the two-arm TOF spectrometer with similar size and intrinsic efficiency of those considered here, we can assume 0.8% as minimum detection efficiency for triple coincidence events. The deduced efficiency is in line with the value obtained in a very recent experiment performed with the TOF-spectrometer CORSET that was arranged for measuring ternary reaction products [54].  
In conclusion, by considering the estimated cross section and detection efficiency, and assuming the use of a beam intensity of 20 pnA (\(1.2\times 10^{11}\) pps) on a \({}^{208}\)Pb target 0.2 \(\upmu\)g/cm\({}^{2}\) thick, about 200 events of \({}^{132}\)Sn + \({}^{68}\)Zn + \({}^{48}\)Ca tripartition can be collected per day. Therefore, in the plots (Figures 13-15), extracted by considering double and triple coincidence modes, 2000 events can be collected in 10 days of beam time provided by existing accelerator facilities such as LNL (Italy) or JYFL (Finland). The cross sections for BF, which are an order of magnitude larger, as well as the larger detection efficiency, makes the collection of thousands of BF events achievable in a few hours with the experimental conditions described above.