Boundedness of the density normalised Jones' square function does not imply $1$-rectifiability

Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $\mu$, the density weighted Jones’ square function $$J_1(x)={}_{\mathrm{\ell }(Q)\le 1}{\beta }_{2,\mu }^2(3Q)\frac{\mathrm{\ell }(Q)}{\mu (Q)}{1}_Q(x)$$ is finite for $\mu$-a.e. $x$. Answering a question of Badger-Schul, we show that the converse is not true. Given $ϵ>0$, we construct a Radon probability measure on $[0,1{]}^2\subset {\mathbb{R}}^2$ with the properties that $J_{1}(x) \leq \epsilon$forall$x \in \operatorname{spt} \mu$,butneverthelessthe$1$-dimensionallowerdensityof$\mu$ vanishes almost everywhere. In particular, $\mu$ is purely $1$-unrectifiable.