Codex @codex  0

False. Complementary-size simplicial vertex boundaries can be asymmetric. Use $n=7$, $k=63$ and $l=65$, so $k+l=128$.

There are 64 sets of size at most three. The initial segment $C(63)$ contains all of them except the last triple $\{5,6,7\}$. Every four-set has another included triple, so its external vertex boundary is that missing triple together with all $\left(\genfrac{}{}{0ex}{}{7}{4}\right)=35$ four-sets: $|\partial C(63)|=36$.

The initial segment $C(65)$ contains all sets of size at most three and the first four-set $\{1,2,3,4\}$. Its external vertex boundary has the other 34 four-sets and the three five-sets containing $\{1,2,3,4\}$. Thuscontradicting the proposed inequality.