  We prove that there exists a finite-dimensional Banach space $X$ such that
$L_1^\mathbb C([0,1])\widehat{\otimes}_\varepsilon X$ fails the strong diameter
two property and $L_\infty^\mathbb C([0,1])\widehat{\otimes}_\pi X^*$ fails to
have octahedral norm. This proves that the octahedrality of the norm
(respectively the strong diameter two property) is not automatically inherited
from one factor by taking projective tensor product (respectively injective
tensor product), which answers [16,Question 4.4].
