Adler's conjecture on spectra of two-cardinal counting functions

Let $T$ be a countable theory and let $f_T(\kappa,\lambda)$ be the associated two-cardinal counting function for infinite cardinals $kappaleqlambda$. Adler's counting-spectrum conjecture. There are only finitely many possible functions $f_T(\kappa,\lambda)$ as $T$ ranges over countable theories, and the property that $T$ is NTP$_2$ can be detected from $f_T(\kappa,\lambda)$. If $T$ has TP$_2$, the function is maximal, namely $f_T(\kappa,\lambda)=\lambda^{\kappa}$ for all $aleph_0leqkappaleqlambda$. The first assertion is addressed by later results in the paper, while the second is explicitly refuted by the counterexample identified below.