As per my another professor and here : http://www.sciencemadness.org/talk/viewthread.php?tid=24831:
According to JACS :
Antiaromaticity only means the planar cyclic structure is less stable than the nearest open ring structure (e.g., cyclopentadienyl carbocation vs. divinylmethyl carbocation). Nothing more than that.
According to JACS, 89, 1112–1119 (DOI: 10.1021/ja00981a015), the cyclopentadienyl carbocation forms easily from cyclopentadienyl alcohol with strong acids even at cryogenic temperatures. The same cannot be said for cyclohexanol. This is good enough indication that the cyclopentadienyl carbocation might be more stable than the cyclohexyl carbocation.
This is a very good and instructive problem. Certainly, I don't know the answer. If for a test, I'd go with your professor. However, the link suggests there may be more to this. I had been skeptical of anti-aromaticity, as it seemed more empirical than theoretical. (I am judging by the use of Frost circles.) I had sought verification of the instability of cyclopentadienyl cation, which as I recall is quite unstable. Anti-aromaticity seemed a good predictor of stability, though I had occasionally found others that were not so sure of its rigor.
It would be my opinion that the rules of aromaticity may be good predictors of compound stability, but as the referred JACS paper indicates, we should remain skeptical of whether a truly repulsive (i.e., high energy) interaction actually occurs. If electron donation to an incipient carbocation were a high energy process as predicted by the rules of aromaticity, then electron donation should be disfavored. The result should be a very slow or no reaction. Assuming the citation is correct, the data indicates a quite different result.
Kudos for the enlightment.