Sharp tangent inequalities in noncommutative $L_p$-spaces

Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semifinite weight, and let $L_p(\mathcal{M})$ denote the associated Haagerup noncommutative $L_p$-space. For $1<p<\infty$, let $q=p/(p-1)$. For $z\in L_q(\mathcal{M})$ and $y\in L_p(\mathcal{M})$, denote by $\langle z,y\rangl