The Krull-dimension conjecture for reciprocal complements

Let $D$ be an integral domain, and let $R(D)$ be its reciprocal complement.

Krull-dimension conjecture. The inequality

$$\dim(R(D)) \leq \dim(D)$$

holds.

This conjecture concerns the relationship between the Krull dimension of an integral domain and that of its reciprocal complement. It was proposed as a general question; the supplied source gives no resolution.