Grothendieck–Witt theory and products of exact categories with duality

Let $I$ be an index set, and for each $i\in I$ let $\mathsf{C}_{i}$ be an exact category with duality. Write $GW(\mathsf{C})$ for its Grothendieck–Witt spectrum.

Product conjecture for Grothendieck–Witt spectra. There is a canonical equivalence of spectra

$$GW\left(\prod_{i}\mathsf{C}_{i}\right)\overset{\sim}{\longrightarrow}\prod_{i}GW\left(\mathsf{C}_{i}\right).$$

The conjecture is motivated by the preceding product decompositions and the behavior of localizing invariants, but the supplied text gives no proof or resolution.