  We initiate the study of a generalization of Kim-independence,
Conant-independence, based on the "strong Kim-dividing" of Kaplan, Ramsey and
Shelah. We introduce an axiom on stationary independence relations essentially
generalizing the "freedom" axiom in some of the free amalgamation theories of
Conant, and show that this axiom provides the correct setting for carrying out
arguments of Chernikov, Kaplan and Ramsey on $\mathrm{NSOP}_{1}$ theories
relative to a stationary independence relation. Generalizing Conant's results
on free amalgamation to the limits of our knowledge of the $\mathrm{NSOP}_{n}$
hierarchy, we show using methods from Conant as well as our previous work that
any theory where the equivalent conditions of this local variant of
$\mathrm{NSOP}_{1}$ holds is either $\mathrm{NSOP}_{1}$ or $\mathrm{SOP}_{3}$
and is either simple or $\mathrm{TP}_{2}$, and observe that these theories give
an interesting class of examples of theories where Conant-independence is
symmetric, including all of Conant's examples, the small cycle-free random
graphs of Shelah and the (finite-language) $\omega$-categorical Hrushovski
constructions of Evans and Wong.
  We then answer a question of Conant, showing that the generic functional
structures of Conant and Kruckman are examples of non-modular free amalgamation
theories, and show that any free amalgamation theory is $\mathrm{NSOP}_{1}$ or
$\mathrm{SOP}_{3}$, while an $\mathrm{NSOP}_{1}$ free amalgamation theory is
simple if and only if it is modular.
  Finally, we show that every theory where Conant-independence is symmetric is
$\mathrm{NSOP}_{4}$. Therefore, symmetry for Conant-independence gives the next
known neostability-theoretic dividing line on the $\mathrm{NSOP}_{n}$ hierarchy
beyond $\mathrm{NSOP}_{1}$. We explain the connection to some established open
questions.
