= Solution
Fill four spatial <molecular orbitals> with the eight $\pi$ <electrons>. For negative $\beta_S,\beta_L$ and $|\beta_S|>|\beta_L|$, the occupied <energies> are $\alpha+\beta_S+\beta_L$, $\alpha+\beta_S-\beta_L$, and the pair at $\alpha-\sqrt{\beta_S^2+\beta_L^2}$. Therefore
$$
E_\pi=8\alpha+4\beta_S-4\sqrt{\beta_S^2+\beta_L^2}.
$$
For equal integrals, $\beta_S=\beta_L=\beta<0$, this becomes
$$
\boxed{E_\pi^{\rm uniform}=8\alpha+4(1+\sqrt2)\beta.}
$$
At equal bond lengths, the last two <electrons> occupy the frontier degeneracy at $E=\alpha$, so their distribution within that degenerate space does not change the bare <Hückel method> <energy>.
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