Political scientists Eric McGhee and Boris Shor have published “Has the Top Two Primary Elected More Moderates?” Anyone may read the 36-page paper at this link. The authors studied California and Washington, which are the only two states that use top-two (as they explain on page 6 and 16, Louisiana and Nebraska don’t have top-two systems).
They conclude that since top-two started in each state, California Democrats in the legislature have become more moderate, but California and Washington Republicans and Democrats in Congress have not; nor have Republicans in the legislatures of either state moderated. The paper suggests that a strong reason why California Democrats in the legislature have moderated is because of three other changes made almost simultaneously: (1) redistricting reform; (2) term limits were eased; (3) the California budget no longer takes a two-thirds majority in each house of the legislature. But the paper believes that top two has probably had some moderating effect on California legislative Democrats.
Page 27 says, “The results of these analyses suggest virtually no effect of the Top Two in Washington or for Republicans in California.” That page also says that because members of congress are not subject to term limits, whereas members of the California legislature are subject to term limits, therefore term limits is probably the main reason why the results are different for California Democratic legislators, versus California Democratic members of the U.S. House. Thanks to Rick Hasen for the link.