You could put the HR mouthpiece for several hours (or even days) into a saturated solution of sodium bicarbonate (baking soda) at room temperature to neutralize the sulphuric acid created by oxidation of sulfur that had been released by degradation of HR and bloomed/escaped onto the surface of the hard rubber. Once submerged into the solution, the mouthpiece will become covered in tiny bubbles of CO2 indicating that neutralization is continuing. Stir the solution from time to time to remove the bubbles and improve the surface contact with the liquid.
Using this method, I was able to remove the sulfuric / acidic taste from a new (but poorly vulcanized) HR mouthpiece. It took about a month until the bubble formation subsided.
I had treated in the same way a different HR mouthpiece that was originally creamy light brown but turned green-brown once washed in cold water. After a 24-hour submersion into the saturated baking soda solution, the green tinge was gone, and the mouthpiece became dark (not light) brown.
It appears that the time necessary for treatment is proportional to the degree of HR deterioration - hours or a few days for light deterioration, multiple days or weeks for strong deterioration. Eventually, the process becomes limited by the rate of migration/diffusion of sulfur from the bulk of HR material onto the surface.
I have not tried it, but I assume that washing soda (sodium carbonate, Na2CO3) saturated solution would have worked faster due to its higher pH compared to sodium bicarbonate (baking soda, NaHCO3).