Sub-additivity of burden in NTP2 theories

In a first-order theory $T$, let $\operatorname{bdn}(a)$ denote the burden of a tuple $a$, and say that $T$ is NTP$_2$ when it has no inp-pattern of depth $\omega$ in a single variable. Sub-additivity conjecture. Burden is sub-additive in NTP$_2$ theories; in particular, for tuples $a$ and $b$, one should have

$$\operatorname{bdn}(ab)\leq\operatorname{bdn}(a)+\operatorname{bdn}(b).$$

Sub-additivity is known for NIP theories and for simple theories, while the conjecture asks whether it holds for all NTP$_2$ theories.