No - there is some right swashplate movement required to tilt the disc forward in order to make the aircraft accelerate BUT that is still forward cyclic.
THEN the dissymmetry of lift caused by differences in induced flow makes the aircraft roll RIGHT - this is the effect which is most noticeable in the transition but is present across the whole speed range and is countered by LEFT cyclic.
Flapback/blowback makes the disc want to rise at the front and that is countered with more FORWARD cyclic (right swashplate movement).
Go and try a transition yourself - from the hover, displace the cyclic forward slightly and hold it - you will see that initially the aircraft descends (if you don't move the collective) then it pitches nose up (flapback) and rolls to the advancing side (inflow roll/transverse flow). It doesn't take a rocket scientist to see that these effects are countered with FORWARD cyclic for flapback and LEFT cyclic for inflow roll (in a helo with an R22 direction of MR rotation).