"Is it true that..." is a difficult question to answer. If you mean, according to a certain school, then yes, according to the Theravada, it is true, since an arahant is unable to practice non-monastic livelihood. They are said to either leave the household life or pass into parinibbana.
See, for example, the enlightenment of Khemā:
At the conclusion of the lesson Khemā was established in Arahatship; the multitude also profited by the lesson.
Said the Teacher to the king, “Great king, Khemā ought either to retire from the world or to pass into Nibbāna.” The king replied, “Reverend Sir, admit her to the Order; as for Nibbāna, never!” She retired from the world and became one of the Teacher’s foremost female lay disciples.
Dhp-A 347 (from Buddhist Legends)
N.B. the word "never" should actually be "enough!", in the sense that it's not time to think about that now.
The point is that, as mentioned in the linked answer, there is a difference between conventional morality and ultimate morality. Conventionally, only those livelihoods mentioned explicitly are wrong, just as only adultery is wrong for one who keeps five precepts. That doesn't mean that ordinary sexual desire is not ultimately misguided; it is. Likewise, any livelihood for the purpose of maintaining a household life is ultimately misguided.
The only exception I can think of is where one's household life is identical to a monastic setting (i.e. no need for ambition, concern for belongings, etc.).