Your translation is single-volitional, not double. 「行こうとしようよ」 means something like "Let's try to try going next week.", which is why it is an incorrect answer.(source)
It seems to be 行こうとする(to try to go) in the volitional form, so it would mean 'let's try to go' and not 'let's try to try to go'.
And since it is incorrect, how would we say "let's try" (with the nuance of ようとする and not with the one of てみる)?