With the exception of my 2008 convertible, all of my convertible Corvettes had hard tops that I installed for cold weather months. Never had a problem with my 56, 66, 91 nor 94 convertible tops being down from October thru April. Had my 66 for 14 years and repaced the convertible top one year before I sold it due to normal weathering. My hard tops were put on at the end of September and removed mid May. The convertible tops were never a problem. The 1994 went thru some terrific rain storms and it never leaked, nor did my 91. I have not heard of any convertible top being damaged while in the down position for the winter months. For my 2008, I store it during the winter months, and since Chevy did not make a hard top for it, I store the Corvette with the convertible top in the up position, but with the front windshield latches loose to prevent stress on the back bow. I could also store the Corvette with the top down, but do not choose to do so. In my experience in the last 50 years with the convertible top and hard top, I've never had a problem. Of course the C1 and C2 convertible tops always leaked since they were not fitted as well as the C4 to C6 convertible tops. Also, I have read posts on this site where owners in the warm southern states hardly ever put the convertible tops up. I have also read that some owners in Arizona and Houston put the hard tops on during the hot summer months and run with the convertible tops stored during the period; car is cooler with the A/C and hard top on. Didn't read any post where the convertible tops were a problem after long storage periods. So my advice is, put the hard top on, don't worry about the convertible top, and enjoy your Corvette. Of course the one disadvantage of installing the hard top is the loss of storage space in the boot because the soft top is there. That's the reason I had luggage racks.
Hope this helps.
Barrett