According to the Pali Theravada suttas (e.g. SN 35.85), the Buddha taught 'sunnata' means 'empty of self & anything pertaining to self'. This includes Nibbana because Nibbana is one of the sense objects (Ud 8.1; ayatana; sense base) included with 'mind objects' in SN 35.85 and because the suttas also say all things are anatta (Dhp 279).
Most importantly, Nibbana is regarded a thing (dhamma) or element (dhatu; MN 115) in the Pali suttas. However, per MN 115 & many other suttas, Nibbana is the unconditioned thing (asankhata dhatu), as follows:
There are these two elements: Dve imā, ānanda, dhātuyo—
the conditioned element and the unconditioned element. saṅkhatādhātu,
When a mendicant knows and sees these two elements, Imā kho, ānanda,
dve dhātuyo yato jānāti passati—
they’re qualified to be called ‘skilled in the elements’.” ettāvatāpi
kho, ānanda, ‘dhātukusalo bhikkhū’ti alaṁvacanāyā”ti.
It follows the Buddha, according to the Pali suttas, never ever not once never ever taught: all phenomena are conditional and empty of any self essence. While Nibbana is empty of any self essence, Nibbana is not conditional.
In summary, it is most essential to understand the verses below:
"All conditioned things [which excludes Nibbana] are impermanent" — when one sees this with wisdom, one turns away from suffering. This is the path to
"All conditioned things [which excludes Nibbana] are unsatisfactory" — when one sees this with wisdom, one turns away from suffering. This is the path to
"All things [which includes Nibbana] are not-self" — when one sees this with wisdom, one turns away from suffering. This is the path to purification.
Also, in the Theravada Abhidhamma, the concept "sabhava" does not mean "own-being". Instead, it means "inherent/own nature", such as in the description below from the Buddha's words:
"And why do you call it 'form'? Because it is afflicted, thus it
is called 'form.' Afflicted with what? With cold & heat & hunger &
thirst, with the touch of flies, mosquitoes, wind, sun, & reptiles.
Because it is afflicted, it is called form.
"And why do you call it 'feeling'? Because it feels, thus it is called
'feeling.' What does it feel? It feels pleasure, it feels pain, it
feels neither-pleasure-nor-pain. Because it feels, it is called
"And why do you call it 'perception'? Because it perceives, thus it is
called 'perception.' What does it perceive? It perceives blue, it
perceives yellow, it perceives red, it perceives white. Because it
perceives, it is called perception.
"And why do you call them 'fabrications'? Because they fabricate
fabricated things, thus they are called 'fabrications.' What do they
fabricate as a fabricated thing? For the sake of form-ness, they
fabricate form as a fabricated thing. For the sake of feeling-ness,
they fabricate feeling as a fabricated thing. For the sake of
perception-hood... For the sake of fabrication-hood... For the sake of
consciousness-hood, they fabricate consciousness as a fabricated
thing. Because they fabricate fabricated things, they are called
"And why do you call it 'consciousness'? Because it cognizes, thus it
is called consciousness. What does it cognize? It cognizes what is
sour, bitter, pungent, sweet, alkaline, non-alkaline, salty, &
unsalty. Because it cognizes, it is called consciousness.
While the Buddha never used the term "sabhava", the text above shows the "inherent nature" of consciousness is to cognise; the "inherent nature" of feeling is to feel; etc.
To say consciousness has no inherent nature appears to say cognition (seeing, hearing, tasting, etc) is not possible.
If consciousness did not have the inherent nature to only cognise, then consciousness would be able to speak, walk, bleed, be reincarnated, etc.
In summary, because consciousness has inherent nature it cannot have 'woke' characteristics, such as 'fluidity'. Consciousness can only be consciousness; consciousness can only cognise. If consciousness had no "sabhava", there would be no sense or life experience; instead; all would be blind, deaf, dumb, etc.