Forcing equivalence of Cohen forcings for non-trivial Fraïssé classes

Let $X$ be the underlying set and let $\AgeK_0$ and $\AgeK_1$ be non-trivial Fraïssé classes, each with at most $|X|$ many elements up to isomorphism. Write $\Cohen[X](\AgeK)$ for the Cohen forcing associated with a Fraïssé class $\AgeK$ on $X$. Forcing-equivalence conjecture. The forcings

$$\Cohen[X](\AgeK_0)\text{ and }\Cohen[X](\AgeK_1)$$

are forcing equivalent. This conjecture extends the corresponding result for countable strong Fraïssé classes and asserts that the countability hypothesis is unnecessary.