Strict increase of a Gaussian norm distribution

Suppose $B$ is a nonzero separable Banach space, the Cameron-Martin space of a Gaussian random variable in a Banach space is a dense subset of $B$, and every centered ball of positive radius has positive probability. Then $t\mapsto \mathbb{P}(\Vert X{\Vert }_B\le t)$ is strictly increasing on $(0,\infty )$. For $0<a<b$, choose $h$ in that Cameron-Martin space of a Gaussian random variable in a Banach space with $\Vert h{\Vert }_B=(a+b)/2$ and $0<\delta <(b-a)/2$. The closed ball of radius $\delta$ centered at $h$ lies in the annulus $a<\Vert x{\Vert }_B<b$ and has positive probability by the symmetric Gaussian translation lower bound. The nonzero assumption is necessary: on $B=\{0\}$ the distribution function equals one everywhere.