The normalized class detects the actual Picard-space map on smooth schemes

The classifying map ${\mathrm{colim}}_m({\mathbb{P}}^m,\mathrm{\infty })\to \mathrm{P}\mathrm{i}\mathrm{c}$ is an A¹-Nisnevich equivalence. PBF after each smooth base change and zero negative weights make the reduced coefficient spectra $F^1({\mathbb{P}}^m)$ constant for $m\ge 1$, in all degrees. Thus the mapping inverse-limit sequence has zero ${\lim }^1{\tilde{\pi }}_1$ term. Restriction of homotopy classes from the suspension spectrum of Picard space to ${\mathbb{P}}^1$ is injective (indeed an isomorphism). These are exactly the steps of [F1].

The hypothesis gives the same reduced class under this restriction, so the spectrum maps from ${\mathrm{\Sigma }}^{\mathrm{\infty }}\mathrm{P}\mathrm{i}\mathrm{c}$ agree. By the suspension-spectrum/Ω∞ adjunction their original Picard-space maps are homotopic. This does not assert a uniqueness theorem for an arbitrary entire spectral map out of $R\mathrm{\Gamma }{\mathbb{G}}_m[1]$ beyond the adjoint Picard-space map needed here.