The only way to verify that claim is by your own senses: flavour and taste in the cup. these are both very subjective, I might be a superficial espresso drinker but for me some good fresh ground beans last at least 3 days if kept sealed and in the fridge.
Ground Coffee has the tendency to absorb every oder odor in the fridge or wherever is kept (in fact one of last uses of stale or unused coffee is has a natural odor eater), so the way it is sealed is very important. Majority of ground coffee seller (Lavazza, Segafreddo, etc.) they vacuum the pack removing almost all the air (check the pack should be hard as a rock before you buy it otherwise the sealing is gone), they claim coffee will stay good even for years - again, it all comes down to what you do with it: if it is moka brew I have had it all my life and most of the times is acceptable, for espresso it is not even ground properly so I never tried.
The only exception to this is Illi sealing, which is packed in nitrogen modified atmosphere I find this ground coffee to be one of the best so far, when you open the pack your do feel the same aroma of fresh ground coffee, I wish they would sell it in an espresso grinding setting (I know it is difficult) to see how it would come out. I regularly buy the decaffeinated version which is ground finer and which I use to make the once-in-a-while-decaf with my oscar.
Last month I started freezing (deep freezing, at least -20) ground coffee inside sealed jars, the result is amazing:
There is no "stale" odor when you open the jar after a week or so: actually no flavor at all, I was expecting it to aggregate into bigger chunks or to develop some kind compactness but no, it is thin as it used to be;
The magic happens when you put it inside a warmed up portafilter, just before pulling the shot: you can feel everything is coming back to life and the espresso is just as it used to be: visually the crema is the same, taste-wise it seems there is more acidity, but I could be wrong and it could be happening only with this variety of beans (100% arabica, almost medium roast)