Some Ashkenazim are medakdek to pronounce their Aramaic "correctly". And they know that there is a mapik heh in Amein Yehei ShemeiH Rabba. Thus,
יְהֵא שְׁמֵ*הּ* רַבָּא מְבָרַךְ לְעָלַם וּלְעָלְמֵי עָלְמַיָּא יִתְבָּרַךְ
But what they don't realize is that, probably due to the phonological features we are trained on, when they try to say Shmēh with a tzeirei (as in English 'hay') they actually say Shmeh with a segol (as in English 'egg'). Listen carefully the next time you or someone else tries it. If it were a different consonant at the end, e.g. a Resh or a Samech, there would be no problem pronouncing the tzeirei, but a difficult guttural like a Heh triggers a weakening into a segol. And it is quite difficult to overcome.
In which case, one is merely substituting one "error" in pronunciation for another. Except you get to sound all weird or precise (depending on one's perspective).
So, which is better? If you don't pronounce the Heh, you sound like other Jews, and one could argue that the meaning is perfectly clear while the standardization legitimizes the 'incorrect' pronunciation. Meanwhile, if you do pronounce the Heh, you are probably still getting it wrong, but just in a vowel, and while at least making an effort to get it right.