SUGRA-SUSY=DM candidate?

Supersymmetry aka SUSY, in a narrow sense, is an extension of the Standard Model, that, like the Standard Model does not include gravity or gravitons or even gravitinos.  The extension of SUSY that includes gravitons and gravitinos is call Supergravity aka SUGRA.

SUSY was formulated because if certain symmetries are present between fermions and bosons in variants of the Standard Model this has a variety of theoretically attractive features.

"Virtual SUSY" approaches.

But, there are hints that the Standard Model may actually have the balance between fermions and bosons that SUSY creates crudely and directly, already in a more subtle way.

For example, there are twelve spin 1/2 fermions (six quarks, three charged leptons and three neutrinos) and twelve spin-1 bosons (the photon, W+, W-, Z and eight gluons) in the Standard Model, both of which seem to be related in a fairly balanced way to the spin-0 Higgs boson mass (which might not need to participate in the fermion-boson symmetry).  Similarly, I've discussed in a previous post a formula in which the Standard Model fermion masses and Standard Model boson masses seem to contribute almost equally to the Higgs boson mass.  And, the "hierarchy problem" which looks at the incredible balancing of huge opposing contributions to the Higgs boson mass that balance out at a tiny electroweak scale value, might seem less unnatural if there are hidden relationships between those masses that relationships like extended versions of Koide's formula suggest.

A more sophisticated variation on this approach looks at Standard Model composite particles called diquarks and mesons as particles that serve the theoretical purposes that superpartners do in a supersymmetry theory.

These approaches are increasingly attractive because a number of the motivations for canonical supersymmetry in which each Standard Model fermion and lepton has a superpartner counterpart are being undermined.  No superpartners or extra Higgs bosons have been found, the Higgs boson mass and a variety of other new discoveries narrow the SUSY parameter space to "unnatural" choices of parameters, unitarity doesn't break down in the Standard Model up to the GUT scale with the current Higgs boson mass, core SUSY dark matter candidates are increasingly disfavored (e.g. by the LUX experiment), neutrinoless double beta decay constraints and proton decay constraints are tightening, and so on.  None of these absolutely rules out some kind of SUSY which has enough moving parts to keep it consistent with experiment for the foreseeable future, but it does take the gleam off it as an attractive theory and favors a closer look at within the Standard Model relationships that can serve the purposes that it was designed to address.

SUGRA-SUSY=Gravitinos

All of this said, even if SUSY isn't necessary because relationships within the Standard Model serve the same purposes, none of this directly addresses the SUSY add on component of Supergravity which adds a graviton and gravitino to the SUSY particle zoo.

The massless spin-2 graviton is by far the hypothetical particle about which there is the most consensus in particle physics.  If there is a quantum gravity of some kind that involves a force carrying boson for gravity of the kind that there is for the other three fundamental forces of the Standard Model, there ought to be a graviton with these properties.

But, it isn't inconceivable that while the other sparticles are unnecessary in physics, that the gravitino, which provides fermion-boson alance in the gravitional sector akin to that in the sector of the other three fundamental forces, could be necessary.  A spin-3/2 particle of the right mass could be a stable or very slowly decaying dark matter candidate, even if the other SUSY particles either don't exist, or only exist at very high energies and are all unstable due to a lack of R-parity conservation.  Particularly in the absence of other sparticles, theoretical limitations on gravitino mass are not great, so a sweet spot like 2 keV-3 keV is not ruled out.  The fact that the SUGRA gravitino comes in just one generation unlike other fermions, and that a gravitino might not be discernable in W and Z boson decays due to conservation of spin, makes it quite attractive as a dark matter candidate even without the remaining SUSY baggage that hatched its existence.

It would also be somewhat reassuring if there were some spin-3/2 fundamental particle in nature, as there are fundamental particles of spins 0, 1/2, 1 and (assuming the graviton exists) 2, but not otherwise of spin-3/2 (there are spin-3/2 baryons, however, which could stand in for a gravitino in a "virtual SUSY" scenario of the kinds discussed above, but wouldn't provide a dark matter candidate, as none of them are even remotely stable) .

Current LHC boundaries on gravitino mass are not a great fit for dark matter.  But, all of those boundaries are model dependent and derive from limitations on other sparticle masses that are excluded by the LHC which are related to gravitino mass in minimial SUGRA models that extend minimal SUSY models.