= Solution
Write $g_s=g_{K_s}$ and let $K_{s,t}$ be the relatively closed hull obtained by mapping the increment $K_t\setminus K_s$ through $g_s$. For a bounded <compact H-hull> $L$, define
$$
\operatorname{rad}(L)=\inf_{x\in\mathbb R}\sup_{z\in L}|z-x|,\qquad \operatorname{rad}(\varnothing)=0.
$$
The <Loewner local growth property> is
$$
\boxed{\lim_{h\downarrow0}\sup_{0\leq t\leq T}\operatorname{rad}(K_{t,t+h})=0\quad\text{for every finite }T.}
$$
One can equivalently use <diameter>: for a nonempty <compact H-hull>, $\operatorname{diam}(L)\leq2\operatorname{rad}(L)\leq2\operatorname{diam}(L)$, using a point of its <closure> on $\mathbb R$ for the second inequality. The condition concerns the mapped increment, not the original difference $K_{t+h}\setminus K_t$, which can be large when a narrow opening closes. It makes growth occur near one continuous <Loewner driving function> rather than by a sudden addition at separated <boundary> points.
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