The first $10^{-35}$ seconds I am a rank amateur, so please forgive me if the answer to this is well-known.
The following quote can in a weekly update for an EdX course I am following in astrophysics:

"And what a week it's been with the recent discovery of primordial gravitational waves from the epoch of inflation!  This one observation has pushed back the earliest direct measurements of the Universe from about 1 second (the epoch of nucleosynthesis) to $10^{-35}$ seconds (the epoch of inflation)."

I would appreciate any comments as to what it will take to close the gap.
EDIT: Well, initially I asked this question with a few specific alternatives. Perhaps - respectfully and with moderators' approval -I should revert to that (as best as I can remember), in that any one of them could be answered with a "yes" or "no" and a few qualifying, suggestive  sentences.
-- Would a viable GUT do the job.
-- Are there currently specifically directed experiments whose intended results will answer the question.
-- Can the gap be closed by correct extrapolation of know experimental and theoretical results.
If this is yet deemed unacceptable, I understand, and please feel free to restrict my question as deemed appropriate.
 A: There exists a huge gap in the strength of the four forces that we have observed in nature between gravity and the other three:




In the following image  we see that the radiation decouples from the "soup" at energy densities of 0.25eV. That is the snapshot of CMB, Cosmic Microwave Background radiation . 

CBR in this plot is the CMB, when we have a snapshot of what happened before then.
The gap you are talking about is covered by known physics up to 10^-11 seconds . The unification of the three stronger forces takes us to the  10^-35 seconds of your question. A unified theory of the three forces with gravity will fill the in between times up to 10^-43 seconds, a huge gap due to the smallness of the gravitational constant. Speculations are listed in the image, and from what I have read the axions are gaining points after the discovery of the imprint of gravitational waves on the CMB. 
A lot of observational and theoretical research is still to be done before any standard model for the universe can be proposed with some confidence.
