"If, while the monk is dwelling by means of this dwelling, his mind inclines to walking back & forth, he walks back & forth [thinking,] 'While I am walking thus, no covetousness or sadness, no evil, unskillful qualities will take possession of me.' In this way he is alert there.
But this is not the case. It is actually said here "while the monk is dwelling ("is living") [generally] by means of this dwelling..." and not "while in the jhana".
Easy to prove. Just look further.
"If, while he is dwelling by means of this dwelling, his mind inclines to speaking, he resolves that 'I will not engage in talk that is base, vulgar, common, ignoble, unbeneficial, that does not lead to disenchantment, dispassion, cessation, calm, direct knowledge, self-awakening, or Unbinding — i.e., talk about kings, robbers, & ministers of state; armies, alarms, & battles; food & drink; clothing, furniture, garlands, & scents; relatives; vehicles; villages, towns, cities, the countryside; women & heroes; the gossip of the street & the well; tales of the dead; tales of diversity, the creation of the world & of the sea; talk of whether things exist or not.' In this way he is alert there. "'But,' [he resolves,] 'I will engage in talk that is scrupulous, conducive to release of awareness,
Elsewhere in other suttas it is said that even in the 1st jhana all thinking (internal speech) completely stops. Not to say about 4th jhana. Even if we don't interpret internal speech as thinking, then this is surely so in the 2nd jhana, where there is no vitakka/vicara at all. So, in the 4th jhana there can be no thoughts and no speech. But one not knowing this, taking as a support this passage from MN 122, may think that in the 4th jhana there is still speech and thinking; and that it can be done while walking -)
Here is okey too, because 4th jhana while walking is also a wrong samadhi -)but this discussion can continue elsewhere, like the thread regarding it here