Cover is probably made from a non breathable fabric, had same problem, apply a coat of Aerospace 303 protectant regularly on tubes and problem solved.

I found a solution that worked for me - simply use VIM with bleach on the tubes - it kills the mold very effectively. 303 never did help me unfortunately. The VIM also has that nice abrasive grit in it to help clean my old tubes - they are not 12 yrs old !