Numbers $k$ with $\{\binom nk:\ n\in\mathbb N\}$ dense in $\mathbb Z_p$ for any prime $p\le k$

Let $$k$$ be a positive integer and let $$p$$ be a prime. In my 2011 PAMS paper joint with my former student W. Zhang [Proc. Amer. Math. Soc. 139(2011), 1569-1577], we studied when $$S(k)=\left\{\binom nk:\ n=0,1,2,\ldots\right\}$$ is a dense subset of the ring $$\mathbb Z_p$$ of $$p$$-padic integers (i.e., $$S(k)$$ contains a complete system of residues modulo any powers of $$p$$).

We call a positive integer $$k$$ universal if $$S(k)$$ is dense in $$\mathbb Z_p$$ for any prime $$p\le k$$. I and W. Zhang showed that $$1,\ 2,\ 3,\ 4,\ 5,\ 9,\ 11,\ 17,\ 29\tag{1}$$ are universal. We believe that there are no other universal numbers.

Question. Can one show that there is no universal number not listed in $$(1)$$?