Let $\mathcal{T}$ and $\mathcal{S}$ be two families of subsets of $[n]$ such that for all $T_i\in \mathcal{T}$ and $S_j\in \mathcal{S}$,

- $|T_i \cap S_j| \neq\emptyset$
- $|T_i| , |S_j| \leq t = O(\log(n))$.

The union of the sets in $\mathcal{T}$ cover all of $[n]$, and likewise for $\mathcal{S}$, and $|\mathcal{T}| + |\mathcal{S}| \leq \mathrm{poly}(n)$. Furthermore, both $\mathcal{T}$ and $\mathcal{S}$ are Sperner families: no $T_i$ is a subset of another $T_j$, and no $S_i$ is a subset of another $S_j$.

Let us say that an element $k\in [n]$ is $\delta$-popular in $\mathcal{T}$ (resp. $\mathcal{S}$) if $k$ occurs in at least a $\delta$-fraction of the sets in $\mathcal{T}$ (resp. $\mathcal{S}$). The properties (1) and (2) above imply that every $T_i\in \mathcal{T}$ contains an element is $(1/t)$-popular in $\mathcal{S}$, and every $S_j\in\mathcal{S}$ contains an element that is $(1/t)$-popular in $\mathcal{T}$. My question is the following:

Is there always an element $k \in [n]$ that is $(1/o(t))$-popular in either $\mathcal{T}$ or $\mathcal{S}$?

For the application that I have in mind I may make one further assumption (which as far as I know, may not be necessary): for every $U\subseteq[n]$, either $T_i\subseteq U$ for some $T_i\in\mathcal{T}$, or $S_j\subseteq [n]\backslash U$ for some $S_j\in \mathcal{S}$.