Hicks: and he will be like himself even when asleep.

Yonge: he will be the same man asleep and awake;

Mensch: He will be himself even when asleep.

Mensch notes DL X.135 (emphasis added):

[135] It is better, in short, that what is well judged in action should not owe its successful issue to the aid of chance. "Exercise thyself in these and kindred precepts day and night, both by thyself and with him who is like unto thee ; then never, either in waking or in dream, wilt thou be disturbed, but wilt live as a god among men. For man loses all semblance of mortality by living in the midst of immortal blessings." Elsewhere he rejects the whole of divination, as in the short epitome, and says, "No means of predicting the future really exists, and if it did, we must regard what happens according to it as nothing to us." Such are his views on life and conduct ; and he has discoursed upon them at greater length elsewhere.

In light of Mensch's note, I would say this characteristic and "armed against fate" are connected.

The definition of the original Greek (and taking DL X.135 into account), to me, implies that the sage is "untroubled by dreams or thoughts of death or predictions of the future."