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This is very interesting. The whole experiment was an attempt to gather information on neutrino oscillations and they find this.
I would assume that the particle never travels faster than c. The question was why and how a neutrino can transform into different particles (electron, muon, and tau). I'd guess the speed/time discrepancy in these experiments would be related to the particle changes themselves. In transforming from an electron to a muon there must be either a superpostioned state where both exist or the wavefunction itself collapses and is 'reborn'. The first is most likely the case. Superpositioned information is not ruled by relativity. "spooky action at a distance".

The question was why and how a neutrino can transform into different particles (electron, muon, and tau).

Neutrinos don't transform into electrons, muons or tauons. According to our experimental observations, and also according to the theoretical Standard Model (i.e., the currently accepted model of elementary particles based on the Weinberg-Salam model), there are three types of neutrinos, each mathematically "partnered" with one of the three basic leptons (i.e., electrons, muons and tauons). Thus there are electron-type neutrinos, muon-type neutrinos and tau-type neutrinos. The neutrinos are all electrically neutral, whereas their partner particles all have non-vanishing charge. Neutrino oscillations occur when a neutrino of one the three types transforms into a neutrino of one of the other types.

Superpositioned information is not ruled by relativity. "spooky action at a distance".

Quantum states that are ordinary superposition states (these are superpositions of so-called "basis" states) exhibit dynamics that are indeed constrained by special relativity (i.e., their dynamical equations are Lorentz invariant). So-called "spooky action at a distance" has to do with what are called "entangled" states - these are special types of superposition states that have the property that they can't be factored into distinct basis states. Their dynamics are also explicitly constrained by Lorentz invariance.

Lorentz invariance is an essential mathematical building block of what is called "quantum field theory" (QFT), which is the principal mathematical tool used to compute predictions regarding elementary particles. There are (literally) millions of verified predictions from experiments carried out over the past 50 years that are all consistent with QFT, and thus consistent with the assumption of Lorentz invariance.

The only experimental evidence ever produced by professional experimental physicists that suggests a violation of Lorentz invariance is the recent announcement regarding the CERN-Gran Sasso "time-of-flight" measurement. If the recently announced measurement is indeed correct, it means that Lorentz invariant QFT is "wrong" in a fundamental way, and needs to be replaced (not just modified, since Lorentz invariance is mathematically embedded everywhere in QFT), and yet, it has somehow produced literally millions of other results that are all nevertheless correct. Thus it is considered by most of us to be highly unlikely that QFT is wrong, and it is more likely that this experimental claim is mistaken.

Most of us suspect that there is a deeply hidden experimental error (since this experiment is comprised of millions of separate experimental parts, and the number of possibilities for error is combinatorically gigantic) but we are waiting with open minds. Spectacular claims need spectacular evidence: this experiment needs to be independently replicated, and then we will see what is what.