I thought I'd compile some of the things I know...

[sheldonison]

So I believe I've managed to prove that the minima/maxima cluster towards the real line; giving arbitrarily close singularities to the real line as the real argument of \( \phi \) grows. And, additionally, that solutions of \( \phi(s) + s = 0 \) necessarily cluster towards the real line...

I'm in the process of cleaning this up; and trying to make the argument a bit more solid than I have it right now....

I look forward to seeing your updated paper.