Not at all. In fact, if I had found a counterexample, it would cause a flurry of new research to quantify exactly how wrong the BSD conjecture is. Such a finding would actually be a boon to their career! That's why my response is curiosity, and not to sneer at them for protecting their extremely secure tenured careers.

Edit 1: And if you think you've found a counterexample to a long-standing conjecture with a computation, you'd better be damned sure that your computation is correct before opening your mouth in public. And that takes a ton of work in the case of the BSD conjecture, because you've almost certainly identified a bug in the extremely complex code underlying that computation. If I ever thought I was holding onto such a counterexample, I'd approach a human calculator like Ralph Greenberg as my first step (after internal checks: re-running the code on another computer to rule out cosmic bit flips, and perhaps running more naive, unoptimized implementations).

Edit 2: This attitude pervades my software development career, and I've brought it to my foray into superconducting circuit design: a bug report brings joy to my life, and I aim to shower the reporter with praise (which may involve chocolate). There is nothing more satisfying than being proven wrong, because it helps us collectively move toward greater truths.