Stability results of octahedrality in tensor product spaces

We prove that there exists a finite-dimensional Banach space $X$ such that $L_1^{\mathbb{C}}([0,1]){\widehat{\otimes }}_{\epsilon }X$ fails the strong diameter two property and $L_{\mathrm{\infty }}^{\mathbb{C}}([0,1]){\widehat{\otimes }}_{\pi }{X}^{\ast }$ 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].