Applied to GS Exec Office Role One month after opening, am I fucked?

Due to an oversight in my tracker I failed to notice that an exec office role had opened, I applied now. I may have a referral, but for FT would that even save me? I don’t think first rounds for GS has started but this late in the process with a referral coming next week potentially do you think it’s salvagable for FT? The role is very niche and small and not advertised in our campus recruting events ( target). I.e my question boils down to whether or not a referall could save my dumbass submitting the app a month later after it was posted for GS FT?