A nice property of $\mathbb P^n$ is:

Property 1: Two subvarieties $U,V$ such that $\text{dim} U +\text{dim} V \geq n$ always intersect.

(for example, any 2 curves in $\mathbb P^2$ intersect)

There are other smooth varieties $X$ when Properties 1 holds. For example, a sufficient condition is that the ranks of $\text{CH}^i_{num}(X)$ are $1$ for $i\leq n/2$. Here $n = \text{dim} X$ and $\text{CH}^i_{num}(X)$ is the Chow group of codimension $i$ modulo numerical equivalences.

My question is whether some converse is true:

Question: Let $X$ be a smooth projective variety satisfying Property 1. Does that impose some upper bounds on the ranks of $\text{CH}^i_{num}(X)$ are $1$ for $i\leq n/2$?

Let's assume we are over $\mathbb C$, but I am also interested in results over any ground fields. One can ask the same questions for the ranks of $\text{CH}^i_{hom}(X)$ (I think they are conjectured to be the same). The baby case is $i=1$, where the question asks if Property 1 tells us something about the rank of the Neron-Severi group of $X$.

I am aware that the question is a little vague (upper bound as function of what?), but that was because of my ignorance, so comments to improve the question are welcome.