Cox ring quotient conjecture for the del Pezzo fibration

Let $\mathcal{F}$ be the projective bundle used to construct the threefold Mori dream space $X$, and let $f$ and $g$ be the defining equations of the two hypersurfaces whose complete intersection is $X$. Cox ring quotient conjecture. The Cox ring of $X$ is

$$\operatorname{Cox}(X)=\dfrac{\operatorname{Cox}(\mathcal{F})}{(f,g)}.$$

This conjecture asserts that the Cox ring of the complete-intersection Mori dream space is obtained by quotienting the Cox ring of its ambient projective bundle by the two defining equations. The supplied text gives no further evidence resolving the conjecture.