Given a topological space $(X,\tau)$, recall that a cover $\mathcal{U}$ of $X$ is locally finite if for every point $x\in \mathcal{U}$ has a neighborhood $U$ that intersects finitely many elements of $\mathcal{U}$.
Instead, we call a cover $\mathcal{U}$ **finitely intersecting** if every member of $\mathcal{U}$ intersect finitely many elements of $\mathcal{U}$.

Recall that $X$ is paracompact if every open cover $\mathcal{U}$ has a locally finite open refinement. Just for the purpose of this question, let us call $X$ **strongly paracompact** if every open cover $\mathcal{U}$ has a finitely intersecting open refinement.

My question is: does this notion coincide with paracompactness?

If the answer is no, I would be very courious to know more about this property. For example:

Has strong paracompactness been studied before, and what is the right name for it?

Is it true that all metrizable second countable spaces are strongly paracompact?

countablymany spines is strongly paracompact. Take such a Hedgehog, and cover it with an $\epsilon$-ball around the joint and countably many open sets, one for each spine. What is the star-finite refinement? $\endgroup$