It seems to me that you are asking a question that is too profound for anyone to dare provide an answer. I will try, but cautiously.
In the way you are formulating your question, it seems you are touching the underlying question of dualism versus monism - i.e. whether there are two "substances" (often represented as mind/body) or one (the physical world is a totality). In general, what you call strong continuity would gravitate toward dualism, whereas monists do also have approaches to that notion which would be in the direction weak continuity.
Also, as the questions noted, you might want to provide a working definition of consciousness for your purposes, especially if you wish to avoid conflicts or confusions with subjects such as psychology.
I also realize that it would be great to have a sensational experiment that would be able to adjudicate between dualism and monism, just as the "double slit experiment" was a major step forward in the particle versus wave theory of light, by showing that photons behave both as waves and particles.
Unfortunately (as you suggested with the EEG experiments), things are not that simple, because we are not dealing here with phenomena that are outside of us and thus measurable and shareable (regardless of the intrinsic difficulties of measuring very small quantities, physical science has grown a remarkable consensus).
By contrast, subjectivity is what it is: subjective. Logical reasoning on this subject are likely to be undecidable, for one specific reason: dualism and monism are conjectures, or more properly axioms, from which a considerable amount of things depend: not only one's view on a specific class of physical phenomena, but on the process of research itself.
About research, we are not dealing merely with experiment data here (a subject on which there is ample consensus) but with considerations on how this process of reasoning with hypothetico-deductive logic and experiments is occurring!
For example, here is a sentence of Henri Poincaré a Mathematician who laid out essential groundwork for the Philosophy of Sciences (at the beginning of his book Science and Hypothesis): "Experience leaves us our freedom of choice, but it guides us by helping us to discern the most convenient path to follow. Our laws are therefore like those of an absolute monarch, who is wise and consults his council of state." While it is a very effective statement, it raises further questions: what is freedom of choice? How can we be an absolute monarch -- who is required to act as an Enlightened ruler?
The problem is that using both logic and experimentation to analyse logic and experimentation is a recursive (feedback) process, a terribly difficult thing to do. It is very easy to end in petitio principii or to derail into unconscious contradictions, or even short-circuits in the mind. And indeed, discussions on this subject sometime degenerate in slanging matches where both logic or experimentation have little to do any longer. It seems we are having a hard time at controlling the feedback process of acquiring data on ourselves ("introspection").
Hence, assuming that someone could make a defining experiment for themselves about consciousness, it could very well be that their inability to make subjective processes objectively observable by others would preclude that knowledge from becoming part of science as we understand it. And discussions might be moot.
I am not saying that the quandary is unsolvable (and there might be several avenues of approach). I am merely laying out why it is a very tough nut to crack.