The BLS signature scheme  promised $b$-bit security for signature with a $2b$-bit appendix, under classical hypothesis leaving aside the possibility of Quantum Computers usable for cryptanalysis. Hovav Shacham's 2005 thesis proposed to use Barreto-Naehrig curves to achieve security comparable to RSA-1024 ($b\approx80$-bit security) with a $2b=160$-bit appendix.
However, per this summary of 2016 attacks, BLS signature using BN curves turned out to require larger appendix; like perhaps $8b/3$-bit appendix for $b$-bit security with $b\approx128$ (with Barreto? and Menezes-Palash-Singh considering $3b$-bit for something conservative).
Is BLS signature still conjectured to achieve $b$-bit security for $2b$-bit appendix using some concrete construction, and which? If not, what comes closest?
If that's not too much a different question, what would be a concrete parametrization for $b\approx128$-bit security?