How do we know that the HSN won't decay before reaching the detector

As neutrinos decay, the flux gets depleted according to:

Where x is the distance travelled and Lambda is the HSN decay length.

We would have to worry about that depletion unless the decay doesn't affect the flux significantly. We can treat the flux as constant if:

That means that the decay length is much larger than the distance travelled (production to decay, roughly 470 m)

At very low mixing angles (what we are probing), decay width is really short. Consequently lifetime and decay length are really long. The number of neutrinos decaying in the detector is REALLY low compared to the actual flux, so the flux can be considered constant while it traverses the detector.

What happens at larger mixing angles? Can HSN decay in a way that affects the flux before even reaching the detector.
We need to worry only up to 1e-7, because other experiments have checked mixing angles larger than that, and they already found nothing.

In the worst case scenario (very high mass, 1e-7 mixing angle), the decay length is still 4 orders of magnitude larger than traversed distance (1000 km!), so we're fine (see attached plot).