100 years? Suspect the generation before your client had the same problem.
Short of a new mortgage to pay dividends (i.e., monetize the illiquid asset - or shares - and, perhaps, exacerbate the problem further for the next generation) it might be best to try to improve upon the basis of the shares by looking at options with the first (most likely) to die and/or a death bed marital gift of the stock? [Not exactly ideal estate planning but it may depend on the life expectancy of the survivor, the exemptions in place at the time for the first to die, state death taxes etc.] At least you might get a more palatable corporate liquidation or sale of shares out of it.
BIG tax wait period was 5 years last I looked but it seems to change by the day. Also assuming election survives estate transfer but not sure on that one.....you definitely need to spend some time looking.