We prove in Section \[Singular\] that (d) is preserved. We do not need (e) to be preserved; we need it only for initial comparison with a barrier function constructed in Section \[Singular\].

Regarding our Closeness Assumptions, we note the following: i) Even for Kähler metrics, condition (d) is necessary so that the $g^2\big(\omega^2{\otimes}\omega^2 + \omega^3{\otimes}\omega^3\big)$ factor in the metric $G$ vanishes before the $\big({{\mathrm}{d}s}{\otimes}{{\mathrm}{d}s}+f^2\,\omega^1{\otimes}\omega^1\big)$ factor does; this is necessary for the singularities to form on ${\mathcal}S^2_-$ (see Theorem 1.1 of [@SW11] and Remark \[kappa-dichotomy\] below). ii) Although we cannot prove that monotonicity of $g$ is preserved for non-Kähler metrics satisfying our Closeness Assumptions, we use condition (e) to prove that for any such metric, $g$ remains above a subsolution we construct that *does* remain strictly monotone in space.

Construction of metrics satisfying the Closeness Assumptions {#sec-initial}
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We choose $f$ to be any smooth function that is defined for $s\in[s_-,s_+]$, is strictly positive except at $s_\pm$, satisfies $|f_s|\leq1$ with equality only at $s_\pm$, and satisfies the closing conditions  and . For each such function, we now construct an infinite-dimensional family ${\mathcal}G_{\alpha,\delta,{\varepsilon}}$ of initial metrics which satisfy our Closeness Assumptions. The family depends on parameters $\alpha,\delta$, and ${\varepsilon}$, to be chosen 