Does a barbarian's rage end if they become fatigued?

[Splendor]

A barbarian rages and then someone hits them with Frostbite (fatigue, no save), does their rage end because they are fatigued?

The reason I ask is because a barbarian's rage is in rounds and not a set duration (10 rounds) it is chosen round by round to keep going and "A barbarian cannot enter a new rage while fatigued or exhausted...". So if something caused him to become fatigued would he be able to continue his rage the following round? If not would he become exhausted? Frostbite says the spell cannot make them exhausted but can another source?

[Chess Pwn]

no, you can become fatigued and still stay in rage, fatigue doesn't stop you from maintaining your current rage, you just can't start a new rage if you're fatigued.

[Onyx Tanuki]

It doesn't say anything in the text for either the baseline or UC barbarian that their rage ends automatically with fatigue, so as long as you can maintain your rage, you shouldn't lose it. You just won't be able to enter a new rage until the fatigue has depleted.

[Sundakan]

Chess is correct. Fatigue prevents you from starting a Rage, but not from continuing it.