The finiteness conjecture for the even Higgs-exponent obstruction set

Let $H_{\mathrm{even}}$ be the set of even integers $m$ for which every prime divisor of $2^m+1$ is 3-Higgs. A prime is 3-Higgs when every prime divisor $q$ of $r-1$ satisfies $v_q(r-1)\leq 3$. The finiteness conjecture. The set $H_{\mathrm{even}}$ is finite. This finiteness would eliminate the remaining even obstruction in the reduction of the unitary-perfect-number problem. The paper proves structural restrictions and finite computational bounds, but explicitly leaves finiteness unresolved.