= Solution
Choose both circular ends of the cylinder to be electric, or rough, boundaries. In the convention of part 1, retain plaquette generators
$$
B_p=\prod_{e\in\partial p}Z_e
$$
with their boundary truncations, and use star generators
$$
A_v=\prod_{e\ni v}X_e
$$
only at vertices not lying on an electric boundary. Omitting the endpoint star checks allows a $Z$ string to terminate at either boundary, which is precisely <anyon condensation at a boundary> for $e$.
Relative homology now has one nontrivial class represented by a primal $Z$ path $\gamma$ joining the two boundaries. The dual absolute homology has one class represented by an $X$ loop $\widetilde\gamma$ around the cylinder. Their strings intersect once and anticommute, so they form one logical pair $\bar Z,\bar X$. Every other closed or boundary-ending string is a product of stabilizers, so the code has exactly one logical qubit.
In the <Random-bond Ising model> mapping, there is no Ising spin for an omitted boundary star. An edge $e$ joining an interior vertex $v$ to an electric boundary is toggled by only $A_v$, so its bond factor $\eta_e\sigma_v\sigma_{v'}$ is replaced by the boundary-field factor $\eta_e\sigma_v$. Equivalently, attach exterior spins fixed to $+1$ and retain the same bond formula. Edges wholly on a boundary contribute fixed constants. The bulk Hamiltonian is therefore supplemented by
$$
H_{\partial}=-J\sum_{e=(v,\partial)}\eta_e(s,q)\sigma_v,
$$
with the signs still determined by $E_s\bar X^q$. This is the required boundary modification.
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