If right eng suffers fire, after fire is extinguished, left eng fails and can't be restarted, will you try to start the right eng considering the urgency?
Yes?
There are some scenarios where I might give consideration to leaving it alone but they would normally involve an assured safe landing. Remember that with jets an engine fire indication doesn’t mean that the wing is engulfed in flames, just that the temperature somewhere inside the nacelle has gone over a pre-defined limit, which can be as low as 250C (IDK what the GE/RR/P&W on the 777 use). It could even be a false warning, although with dual loops this is much less likely.
Considering the remote and/or unlandable areas a LH aircraft like the 777 often flies over, the chances of surviving the dead-stick landing/ditching *and* the subsequent waiting around for someone to rescue you could be fairly slim. Even if the restarted engine gives more warnings, as long as it produces thrust I’d be minded to keep it running while I headed for an alternate...