Existence of an integral domain with non-plethory integer-valued polynomial ring

Let $D$ be an integral domain. Non-plethory-domain conjecture. There exists an integral domain $D$ such that ${\operatorname{Int}}(D)$ is not an (idempotent) $D$-plethory. The preceding results establish analogous failures for certain rings with zero divisors, while the conjecture asks whether such a failure occurs for an integral domain.