  We show that a very general (3,3) complete intersection in $\mathbb{P}^7$
over an algebraically closed uncountable field of characteristic different from
2 admits no decomposition of the diagonal, in particular it is not retract
rational. This strengthens Nicaise--Ottem's result, where stable irrationality
in characteristic 0 was shown. The main tool is a Chow-theoretic obstruction
which was found by Pavic--Schreieder, where quartic fivefolds are studied.
