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The TOTEM experiment at the LHC has performed the first measurement at$\sqrt{s} = 13$ TeV of the $\rho$ parameter, the real to imaginary ratio of thenuclear elastic scattering amplitude at $t=0$, obtaining the following results:$\rho = 0.09 \pm 0.01$ and $\rho = 0.10 \pm 0.01$, depending on differentphysics assumptions and mathematical modelling. The unprecedented precision ofthe $\rho$ measurement, combined with the TOTEM total cross-sectionmeasurements in an energy range larger than 10 TeV (from 2.76 to 13 TeV), hasimplied the exclusion of all the models classified and published by COMPETE.The $\rho$ results obtained by TOTEM are compatible with the predictions, fromalternative theoretical models both in the Regge-like framework and in the QCDframework, of a colourless 3-gluon bound state exchange in the $t$-channel ofthe proton-proton elastic scattering. On the contrary, if shown that the3-gluon bound state $t$-channel exchange is not of importance for thedescription of elastic scattering, the $\rho$ value determined by TOTEM wouldrepresent a first evidence of a slowing down of the total cross-section growthat higher energies. The very low-$|t|$ reach allowed also to determine theabsolute normalisation using the Coulomb amplitude for the first time at theLHC and obtain a new total proton-proton cross-section measurement$\sigma_{tot} = 110.3 \pm 3.5$ mb, completely independent from the previousTOTEM determination. Combining the two TOTEM results yields $\sigma_{tot} =110.5 \pm 2.4$ mb.

First determination of the $\rho $ parameter at $\sqrt{s} = 13$ TeV -- probing the existence of a colourless three-gluon bound state