Quoting off-channel communication with @daveshah:

There are really two GSR signals. The first one is an internal one driven by configuration logic that resets all flip flops at startup (perhaps better called PUR).
The second one is the user GSR signal which can optionally be used to reset flip flops. This is what GSR/SGSR drive.
However, right now Yosys and nextpnr tend to set GSR=DISABLED on everything so GSR/SGSR would have no effect
I've got way too much backlog to deal with this any time soon
Hmm, looks like nextpnr should already support SGSR if you use the Lattice postpnr primitive (which is how Trellis represents it) - a GSR primitive with CLK and GSR pins, and the "SYNCMODE" parameter set to "ASYNC", so it would only need a small packer rule to support SGSR itself (edited)
But as above it wouldn't do anything without GSR being enabled on flipflops etc

Upstream is working on this: YosysHQ/yosys#1332 YosysHQ/nextpnr#319

Merged upstream.

Needs YosysHQ/yosys#1343 upstream.

Merged.