Spectral GRID invariants and Lagrangian cobordisms

We prove that the filtered GRID invariants of Legendrian links in link Floer homology, and consequently their associated invariants in the spectral sequence, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on ${\mathbb{R}}^3$, strengthening a result by Baldwin, Lidman, and the fifth author.