I would play the red circled notes as F#s because there's an F# tied over from the previous bar. You then get an F# tied over to the next bar, that's why you have the natural sign for the next F. Basically play it as it is, I can see what they mean.
However -
I just did an experiment typing the same notes into Sibelius, and it recognised the first tied note as being an F#, but it wanted another # sign by the next F, it didn't automatically make it a sharp, and wouldn't let me just delete the natural sign.