Suppose that a topological space $X$ satisfies the following property

(P): "Each uncountable disjoint open family is locally countable",

where a family $\mathcal U$ of subsets of $X$ is called locally countable if each point of this space has a neighbourhood that intersects at most countably members of $\mathcal U$.

We do not know whether this property have been studied before.

If yes, what this property is called? Any reference/suggestion would be a great help for us.