Is the Berman-Hartmanis Conjecture Solved? The Berman-Hartmanis conjecture more or less states that if one-way functions exist, there are some problems in $NP$ which cannot be polynomially reduced to $NP$-complete (cf. Ker-I Ko, A Note on One-Way Functions and Polynomial-Time Isomorphism).

Since the Cook-Levin theorem states that $NP$ is polynomial-time reducible to $NP$-complete.

Does the Berman-Hartmanis conjecture state anything about the converse statement? I.e. if $NP$ is polynomial-time isomorphic to $NP$, then one-way functions do not exist? 

It seems to me that proving the converse and proving the non-existence of one-way functions proves the $P=NP$ problem.

Furthemore, is the Berman-Hartmanis conjecture solved?
The conjecture remains open. Also there is a proof that shows that if the conjecture is true, then P=/=NP. To my knowledge no one has proven the opposite, that if the converse is true then P=NP, to be true.