Is it possible to have an $\omega_2$-length sequence of ($\omega_1$-)Suslin trees such that if one builds the product of finitely many trees in that sequence, one ends up with a Suslin tree again?

The existence of such a sequence of length $\omega$ follows from $\diamondsuit$, as was shown by Jensen. By Shelah and independently Todorcevic, already a Cohen real gives rise to a Suslin tree, so it could be possible that a adding $\aleph_2$ many Cohen reals produces such a sequence.