Oura's conjecture on analogies between Eisenstein series and Eisenstein polynomials

Oura introduced Eisenstein polynomials $\varphi_\ell^{H_1}(x_0,x_1)$ associated with the group $H_1$, and, when this polynomial is nonzero, let $\widetilde{\varphi_\ell^{H_1}}$ denote it divided by its $x_0^\ell$ coefficient. Let $Th$ denote the transform used to associate a univariate polynomial to this normalized Eisenstein polynomial. For an odd prime $p$, a rational number $a$ is $p$-integral when $v_p(a)\geq 0$, where $v_p$ is the valuation on $\mathbb{Q}$. The relevant Eisenstein-series properties concern zeros on the arc $\{e^{\sqrt{-1}\theta}\mid \pi/2\leq\theta\leq2\pi/3\}$, equality of zeros after increasing the index by the corresponding weight step, and $p$-integrality of coefficients.

Oura's conjecture. The analogous properties hold for the transforms of the normalized Eisenstein polynomials, namely:

1. All zeros of $Th(\widetilde{\varphi_\ell^{H_1}})$ lie on the circle $\{e^{\sqrt{-1}\theta}\mid \pi/2\leq\theta\leq2\pi/3\}$.
2. The zeros of $Th(\widetilde{\varphi_\ell^{H_1}})$ are the same as those of $Th(\widetilde{\varphi_{\ell+4}^{H_1}})$.
3. If $p$ is an odd prime, then the coefficients of $Th(\widetilde{\varphi_{2(p-1)}^{H_1}})$ are $p$-integral.

These claims are proposed as analogues of known properties of Eisenstein series, connecting the zero distributions, index-shift behavior, and arithmetic integrality of the two theories. The supplied text does not state that any of the three claims has been proved or disproved.