That is not at all a stretch and is slightly offensive that you used the word “stretch” here given what that word is supposed to mean. This is a perfectly valid concern where many people will be pressured into complying.
Except if you don’t take the loan, you are still betting big on the company by staying around for stock options. There is zero reason to think this shows lack of faith in the company. This is on the level of thinking you’re going to get fired if you don’t buy a red BMW because all the execs have red BMWs and got a local dealer to offer a good deal to all the employees.