The day seems highly unlikely... Just before the election, going against WH7 in an already packed month and with just 3 weeks of promo? No way.
Looking at the schedule:
[2012.04.25] Itano Tomomi - 3rd Single "10nen go no Kimi e"
[2012.05.02] Nogizaka46 - 2nd Single "Oide Shampoo"
[2012.05.02] Sashihara Rino - 1st Single "Soredemo Suki da yo"
[2012.05.09] NMB48 - 4th Single "Nagiichi"
[2012.05.16] SKE48 - 9th Single "Aishite Raburu!"
[2012.05.23] AKB48 - 26th Single "Manatsu no Sounds Good!"
[2012.05.30] Watarirouka Hashiritai 7 - 10th Single "Shounen yo Uso wo Tsuke"
[2012.06.06] The 4th Senbatsu Election
[2012.06.13] Ono Erena - 1st Single "Erepyon" [2012.06.20] Maeda Atsuko - 2nd Single "Kimi wa Boku da"
[2012.07.11] French Kiss - 5th Single "TBA"
I'd say 06-27, 07-04 or 07-18 is far more likely.
and yay, new NY