So does an unconscious being has a mind if there is no cognition whatsoever?
Yes, it continues as the ālayavijñāna, for example, in nirodha-samapatti.
Isn't it the case that for an untrained being, he is unconscious at the time of black near attainment? If yes, can an unconscious being at the time of black near attainment has luminosity as the object of his mind?
It is not that an untrained being is unconscious. He or she is simply not able to recognize the moment of luminosity when it occurs because it is very brief.
Since you held earlier that the two truths are only cognitions, then in the above two instances, there can be no two truths since there are no cognitions.
This implies that the two truths as cognitions is subjective. Therefore your positing of two truths as cognitions is implicitly in a framework that is not free from subject-object duality.
Well, the way Candrakirti defines the two truths is a little more precise: he states that the two truths are the object
of either true or false cognitions respectively. In this case then, luminosity would be an ultimate truth. However, when luminosity is not correctly perceived, it becomes a relative truth; when it is correctly perceived, it is an ultimate truth.
The point however is that the truths are defined on the basis of veridical or false cognitions, so as a shorthand, I place the emphasis on the cognitions since they are the defining factor.
So yes, the two truths are conventionally subjective, they are not objective. The object of a veridical cognition, dharmatā śunyatā, is the ultimate truth. Nevertheless, the realization of ultimately truth comes when one's mind is truly synchronized with how things are (yatha bhutaṃ) and the apparent duality of subject and object vanishes since it is not there to begin with.
While one is confined to relative truth cognitions (including relative formulations of ultimate truths) one is necessarily confined to subject/object duality. There is no conceivable way to avoid this as long as we are using words and concepts.