We present observations of the molecular gas in the nuclear environment of three prototypical low-luminosity active galactic nuclei (LLAGNs), based on VLT/SINFONI AO-assisted integral-field spectroscopy of H2 1-0 S(1) emission at angular resolutions of ~0.''17. On scales of 50-150 pc, the spatial distribution and kinematics of the molecular gas are consistent with a rotating thin disk, where the ratio of rotation (V) to dispersion (σ) exceeds unity. However, in the central 50 pc, the observations reveal a geometrically and optically thick structure of molecular gas (V/σ < 1 and N H > 1023 cm–2) that is likely to be associated with the outer extent of any smaller scale obscuring structure. In contrast to Seyfert galaxies, the molecular gas in LLAGNs has a V/σ < 1 over an area that is ~9 times smaller and column densities that are on average ~3 times smaller. We interpret these results as evidence for a gradual disappearance of the nuclear obscuring structure. While a disk wind may not be able to maintain a thick rotating structure at these luminosities, inflow of material into the nuclear region could provide sufficient energy to sustain it. In this context, LLAGNs may represent the final phase of accretion in current theories of torus evolution. While the inflow rate is considerable during the Seyfert phase, it is slowly decreasing, and the collisional disk is gradually transitioning to become geometrically thin. Furthermore, the nuclear region of these LLAGNs is dominated by intermediate-age/old stellar populations (with little or no ongoing star formation), consistent with a late stage of evolution. Based on observations at the European Southern Observatory VLT (082.B-0709 and 084.B-0568).