Is the Picard-to-operad Picard homomorphism an isomorphism?

Determine whether the natural homomorphism Pic(LSp, ∧) → Pic(Operad(LSp), ∧) that sends an invertible spectrum X to the operad CoEnd(X) is an isomorphism; in particular, resolve the case LSp = Sp.

Background

In the paper of Morita theory for operads, the authors consider the symmetric monoidal category of operads (Operad(LSp), ∧) and a natural homomorphism from the Picard group of (LSp, ∧) to the Picard group of (Operad(LSp), ∧) given by X ↦ CoEnd(X). Whether this homomorphism is an isomorphism has implications for classifying operads up to Morita equivalence and for understanding how invertible spectra act on operadic structures.

The footnote explicitly flags the status of this homomorphism as an open question even in the classical setting LSp = Sp, indicating a gap in current knowledge about the relationship between invertible spectra and invertible operads under the smash product.