Oura's analogy conjecture for Eisenstein polynomials

Let Eisenstein series and Eisenstein polynomials be the corresponding invariant-function constructions for the groups described in the paper. For a nonzero polynomial $\varphi_\ell^{G_{\rm{II}}}(x,y)$, let $\widetilde{\varphi_\ell^{G_{\rm{II}}}}(x,y)$ denote the polynomial divided by its $x^\ell$ coefficient, and let $Th$ be the transformation defined in the paper. For an odd prime $p$, say that a rational coefficient is $p$-integral when its $p$-adic valuation is nonnegative.

Oura's conjecture. The following analogous properties hold for $Th(\widetilde{\varphi_\ell^{G_{\rm{II}}}})$: (1) all zeros lie on a segment of the circle $\{e^{\sqrt{-1}\theta}\mid \pi/2\leq\theta\leq2\pi/3\}$; (2) its zeros are the same as those of $Th(\widetilde{\varphi_{\ell+8}^{G_{\rm{II}}}})$; and (3), for every odd prime $p$, the coefficients of $Th(\widetilde{\varphi_{2(p-1)}^{G_{\rm{II}}}})$ are $p$-integral.

This conjecture proposes that three characteristic properties of Eisenstein series—zero location, periodicity of zero sets, and integrality of coefficients—also occur for the associated Eisenstein polynomials. The supplied text gives no resolution status.