The Skolem property in rings of integer-valued polynomials

Let $D$ be an integral domain with quotient field $K$ and $E\subseteq D$. We investigate the relationship between the Skolem and completely integrally closed properties in the ring of integer-valued polynomials

Among other things, we show for the case $D=\mathbb{Z}$ and $\vert E \vert = \infty$ that the following are equivalent: (1) $\text{Int}(E,\mathbb{Z})$ is strongly Skolem, (2) $\text{Int}(E,\mathbb{Z})$ is completely integrally closed, and (3) $\text{Int}(E,\mathbb{Z})= \text{Int}(E\backslash\{a\}, \mathbb{Z})$ for every $a\in E$.