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The negative frequency-dependent dielectric constant is always found in the GW calculation. For example:
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freq. eps_1 eps_2
0.000 8.342 0.000 dielectric constant
0.934 8.684 -0.572 dielectric constant
1.865 9.680 -2.037 dielectric constant
2.793 9.263 -5.241 dielectric constant
3.715 6.544 -7.208 dielectric constant
4.630 4.004 -6.994 dielectric constant
.....
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In GGA and Hybrid functional calculations, those epsilon_2 are positive as they should be. The following is the GW-only INCAR:

# general:
SYSTEM = XXX
ISTART = 1
ENCUT = 300.0
ISMEAR = 0
SIGMA = 0.01
LMAXMIX = 4
ADDGRID = .TRUE.
PREC = accurate
LORBIT = 11

# GW
LMAXFOCKAE = 4
ALGO = GW0 # algorithm to setup response function
ENCUTFOCK = 0
NELM = 1 # number of electronic SC steps
NKRED = 2
NBANDS = 320
NOMEGA = 64

# LHFCALC = .TRUE.
LSPECTRAL = .TRUE.
ENCUTLF = 200 # plane wave cuoff in local-field corrections
ENCUTGW = 200 # plane wave cutoff in response function chi

The above problem is very general, not specific to one system. Any idea? Thanks!
<span class='smallblacktext'>[ Edited ]</span>

did you get any update on this?