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 R 3 , strengthening a result by Baldwin, Lidman, and the fifth author.1 This contrasts with the situation in smooth topology, where a cobordism exists between any pair of links, but the minimum genus of such a cobordism is difficult to determine.2 In particular, this also illustrates the important fact that exact Lagrangian cobordisms are directed.