Counting-types characterization of NTP$_2$

Let $\varphi(x,y)$ be a partitioned formula, and let $f_{\varphi}(k,l)$ denote the number of pairwise-inconsistent partial $\varphi$-types of size $k$ over a parameter set of size $l$. For $tin\boldsymbol{N}$, let $\bigwedge_tvarphi$ denote the associated $t$-fold conjunction formula. Counting-types conjecture. The following are equivalent: (i) $\bigwedge_tvarphi$ is NTP$_2$ for every $tin\boldsymbol{N}$; (ii) $f_{\bigwedge_tvarphi}(k,l)$ is polynomially bounded for every $tin\boldsymbol{N}$. At minimum, $f_{\varphi}(k,l)$ should be polynomially bounded for every formula in an NTP$_2$ theory. The paper proves that polynomial boundedness follows from ULCFS and obtains an $\varepsilon$-power-saving result, but the proposed characterization is open.