Analogues of the Berman Hartmanis conjecture and the Creativity Hypothesis
First of all, Mahaney's Theorem says that merely assuming $\mathsf{P} \neq \mathsf{NP}$, there are no sparse $\mathsf{NP}$-complete sets. (Historically, Mahaney was motivated to study this precisely because of Berman-Hartmanis, but the theorem is independent of BH Isomorphism.)

The $\mathsf{AC}^0$ version of the isomorphism conjecture (with various restrictions on uniformity) is a theorem. See the following paper and references therein:


  Manindra Agrawal, The Isomorphism Conjecture for Constant Depth Reductions, JCSS 77 (special issue on Karp's Kyoto Prize): 3-13, 2011.


You may also be interested in other work of Agrawal, Allender, and others on related questions. Agrawal's and Allender's webpages have links to most of the relevant papers (I only say "most" in case I've missed a few that didn't involve either Agrawal or Allender, but I haven't done a thorough literature review recently).

On creativity, Joseph and Young's construction is still one of the main pieces of evidence against BH.