Combinatorial Properties Related to the Higher Baumgartner's Axiom

We isolate two combinatorial properties, each expressible by a $\Pi_2$-sentence over the structure $(H(\omega_3),\in,\omega_1,\omega_2,\text{NS}_{\omega_2})$, such that each property is consistent with CH, and their conjunction together with $2^\omega \le \omega_2$ and $2^{\omega_1} = 2^{\omega_2} = \omega_3$ implies the existence of a c.c.c. forcing which forces the higher Baumgartner's axiom.