I bought a 1/4" x 3" ruby rod some years ago and noticed that when I pushed one end against an ice cube the rod became cold in a very short time. As ruby is basically sapphire with a tiny bit of chromium added, it has a similar thermal conductivity to sapphire. Turns out sapphire and ruby have thermal conductivities nearly the same as pure silver:
Silver ....... 0.974 cal/cm-s-K (407.8 W/m-K)
Sapphire ... 0.955 cal/cm-s-K (399.8 W/m-K) -- 98.0% of silver
Copper ..... 0.923 cal/cm-s-K (386.4 W/m-K) -- 94.8% of silver
Gold ......... 0.7003 cal/cm-s-K (293.2 W/m-K) -- 71.9% of silver
Try this experiment with one of your smaller pieces. It's amazing to see how fast it gets cold.