It appears that results have been released on the first Tuesday of the month for the past few years, so I predict that the results will be released on April 3. They also sometimes come out on Fridays, so my second guess is April 6, which is the Friday of the same week.
Also, here's a frequentist estimate that ignores the day of the week. Among other invalid assumptions, we assume that the last six years are a random sample of years,
#!/usr/bin/Rscript
release_dates_text <- c('Apr 17', 'Mar 23','Apr 1', 'Apr 10', 'Apr 6', 'Apr 5')
release_dates <- as.POSIXct(release_dates_text, format = '%b %d')
conf.int <- as.POSIXct(t.test(as.double(release_dates))$conf.int[1:2], origin = '1970-01-01')
print(mean(release_dates))
print(conf.int)
and we get a 95% confidence interval of the population mean of March 27 to April 14 (much wider than the credible interval that qlathrop estimated). And if we had to guess based on this analysis, we'd say April 5.
But again, this estimate is stupid because it doesn't consider domain knowledge; considering day of the week, I predict that the results will be released on April 3 or maybe April 6.