The rubber smell/taste (mostly an aroma, correct?) is usually not from contact with a stopper. Regarding the staying power of the objectionable smell, well, it may go away with time... or it may not. It really depends on exactly what you're smelling. Most of the rubbery aromas in fermented beverages come from reduced sulphur compounds (remember that hydrogen sulphide, H2S, is often produced by yeast that are not sufficiently nourished - deficiencies in yeast assimilable nitrogen will cause the yeast to generate it). While the "rotten egg" aroma of sulphides is well known, the good news is that with a little aeration of your must late in fermentation, most of it will come out of solution. The technique called "splash racking" is often used to clear hydrogen sulphide smells.
However, if the sulphide is left in solution for any length of time (more than a few days), in the presence of ethanol, hydrogen sulphide will chemically change. It frst becomes a class of compounds called mercaptans (sometimes called mono-mercaptans), and eventually becomes completely reduced sulphur compounds, called disulphides. Mercaptans will react with free copper ions in a liquid to form a compound which is chemically more stable, not readily soluble in water, and it will precipitate out, clearing the problem and removing the odor. So often the "copper penny trick" is employed to introduce some free copper into the solution - that's where you drop a clean penny, or a clean piece of copper tube, or even a copper pot scrubber into the must and allow it to stay a day or two until the aroma clears - or not.
Sometimes instead of introducing copper metal, some winemakers will add a carefully measured amount of copper sulphate solution because that will have the same effect. But the sulphate solution by itself may not work, either.
You see, while mercaptans will react with the copper in this way, disulphides will not. You need to oxidize any disulphides first (turning them back into mercaptans) before the copper trick will work. A recognized safe way to turn disulphides back into mercaptans is to add a little ascorbic acid in addition to the copper, to the must. It will slowly (over a month or longer in many cases) change the disulphides into chemicals that will react with the free copper in solution.
BUT working with copper solutions in wine or mead is inherently dangerous, since too much copper is toxic. Although the copper penny or tube trick generally doesn't add enough copper to be a problem unless you leave the metal in there for longer than a few days, I especially
DON'T recommend the copper sulphate addition method unless you are experienced with chemical lab procedure and you can accurately measure reagents and titrate solutions. It is
VERY EASY to add too much copper via sulphate additions to your mead, rendering it toxic - copper poisoning is similar to lead or mercury poisoning, so in addition to internal organ damage you can also develop brain and nervous system problems that can really mess with your head.
Given that the smell/taste of methyl mercaptan resembles that of vulcanized rubber (aka your stoppers), that may be the predominant chemical in your meads at present. If that is true, since the mercaptans react readily with free copper, the penny trick may work for you without having to resort to ascorbic acid additional treatment. You can try it, and let us know how it goes.
OK, Medsen, this time you beat me to it!