"To continue the speculation" means to keep discussing or guessing about something without having definite information or confirmation. Full definition
There's continued speculation on where the game might fit within the timeline that we know. (gonintendo.com)
Brown was mentioned as a candidate for Lt. Governor back in 2010 and, with continued speculation over Duffy's future, his name keeps coming up as a potential addition to the 2014 ticket. (nystateofpolitics.com)
Striker Papiss Cisse remains committed to Newcastle amid continuing speculation over his future on Tyneside. (itv.com)