Having inherited a stupidly large rock that had been in my family since 1905,I moved immediately to divest, and it wasn't in any way hard to sell, just really costly.
Even with step-up cost basis cancelling out huge capital gains, Sothebys bit off about 22% in commissions and fees for itself.
I also explored selling through an individual jeweler, and was offered 6 figures cash in a briefcase, but the commission would have been about the same, just more camouflaged, and I had no reason to want to explain a six-figures-of-cash deposit to my bank I'd have had to make thereafter.
Big high quality diamonds at least, are liquid these days. (Your link is from 1982!) Just expect considerable slippage when you sell, the inevitable middlemen have sharp teeth.