LQRE and quasi-strict equilibrium

A not-terribly-standard but conceivably interesting refinement of Nash equilibrium is "the rational limit of logistic quantal response equilibria".  Last night I had myself convinced that it was related to quasi-strict equilibrium; if there is a relationship between the two, though, it's complicated. The simplest precise conjectures that I don't know to be false are (1) any quasi-strict NE has the same payoff as some RLLQRE and (2) any connected set of quasi-strict NE includes a RLLQRE.  It may also be that there is a tighter relationship in 2 player games than more generally; in particular, every game has an RLLQRE, and every 2-player game has a quasi-strict NE, but not every larger game has a quasi-strict NE; it wouldn't surprise me if every RLLQRE in two-player games is quasi-strict, but that obviously can't be true in games that lack a quasi-strict equilibrium.