Non-flatness conjecture for rings of integer-valued polynomials

Let $D$ be an integral domain, and let $\operatorname{Int}(D)$ denote its ring of integer-valued polynomials. Consider the two local domains

$$D=\mathbb F_2[[T^2,T^3]]$$

or

$$D=\mathbb F_2+T\mathbb F_4[[T]].$$

Non-flatness conjecture. The domain $\operatorname{Int}(D)$ is not flat over $D$ for either of these choices of $D$; in particular, there exists a local, Noetherian, one-dimensional, analytically irreducible integral domain $D$ such that $\operatorname{Int}(D)$ is not flat over $D$.

The conjecture would provide examples showing that rings of integer-valued polynomials need not be flat, complementing the source's discussion of cases where $\operatorname{Int}(D)$ is locally free and flat. The supplied source gives no resolution evidence.