Characterising 1-rectifiable metric spaces via connected tangent spaces

We prove that in a complete metric space $X$, $1$-rectifiability of a set $E\subset X$ with ${\mathcal{H}}^1(E)<\mathrm{\infty }$ and positive lower density ${\mathcal{H}}^1$-a.e. is implied by the property that all tangent spaces are connected metric spaces.