Just got a new personal best on Infinite Descent (a sub 4:10, screenshotted) and when I went into the app to look it up in the stats, I found that
a) SRL doesn't have it's own heading in the activities and
b) Going into recent activities does accurately tell you where you were and who you raced against, but only gives you kills, deaths and kdrs, not race times.
I know that technically, SRL is a crucible mode, but deaths are unimportant (apart from taking valuable time away) and kills are something you get as a byproduct or chase for the bounty, and I think the stat readout on the app should be adjusted to show the race times and maybe kills as a fun 'I turned this many enemies into roadkill' sidenote.
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If you want stats you have to buy the $10 logbook at the Eververse store. When completed it gives you a special set of race gear and a sparrow and shader to match.