Künneth transversality conjecture for motives

Let $f_1:X_1\to Y_1$ and $f_2:X_2\to Y_2$ be morphisms of schemes, and let

$$f:X_1\times_k X_2\to Y_1\times_k Y_2$$

be their product. For $i=1,2$, let $F_i\in\mathbf{T}(X_i)$, and let $F_1\boxtimes F_2$ denote their external product. Künneth transversality conjecture. If $f_i$ is $F_i$-transversal for $i=1,2$, then $f$ is $F_1\boxtimes F_2$-transversal. This statement appears in the supplied text as a proposition and is therefore treated as established rather than an open conjecture.