A map with small image on a whole star has a common hole with every map (Erdős 624)

Let $U$ be a finite set with $n=2^k$ elements. For a map $g:\mathcal P(U)\to U$ and $Y\subseteq U$ write $I_g(Y)=\{g(A):A\subseteq Y\}$. For $B\subseteq U$ let $\mathcal K_B$ be the set of $k$-subsets $Y\supseteq B$.

Theorem. Let $k\ge3$, $S\subseteq U$ with $|S|\le k-2$, and let $h:\mathcal P(U)\to U$ satisfy $64|I_h(Y)|\le n$ for every $Y\in\mathcal K_S$. Then for every map $g:\mathcal P(U)\to U$,

In particular $h$ and $g$ have a common hole on some $k$-set through $S$. Context: Erdős Problem 624 asks whether $H(n)-\log_2 n\to\infty$, where $H(n)$ is the least $m$ such that some $f:\mathcal P(X)\to X$ ($|X|=n$) satisfies $f[\mathcal P(Y)]=X$ for all $|Y|\ge m$. By an earlier track-amplification result it would suffice to show that, for large $k$, any two maps $\mathcal P(U)\to U$ with $|U|=2^k$ have a common hole: a $k$-set $Y$ and a point $u$ missed by both maps on $\mathcal P(Y)$. These results do not settle Problem 624.

Formalization Note. Counting form with natural numbers (truncated subtraction is harmless since $|I_g(Y)|\le n$); $U$ is an arbitrary Fintype with Fintype.card U = 2 ^ k.