The double-dual isomorphism for finite tensor categories

Let ${\mathcal C}$ be a finite tensor category, let $V\in{\mathcal C}$, let $V^{**}$ denote the double dual of $V$, and let $L_\rho$ be the distinguished invertible object of ${\mathcal C}$. Double-dual conjecture. There exists a natural isomorphism of tensor functors

$$V^{**}\to L_\rho^*\otimes {}^{**}V\otimes L_\rho.$$

This asserts that the double-dual functor is related naturally to the corresponding double-dual operation by conjugation with the distinguished invertible object; the supplied text gives no resolution status.