Meagreness of the non-tangential singular set

Let $\mu$ be a measure under consideration, and let $\mathcal{S}_{nt}^{sing}(\mu)$ denote its non-tangential singular set. Meagreness conjecture. The set $\mathcal{S}_{nt}^{sing}(\mu)$ is meagre in $\mathbb{R}$.

This claim is resolved according to the supplied status evidence, which points to a lemma constructing a measure $\mu\in\mathcal{M}_{c,1}^{\lambda}(\mathbb{R})$ with $\lambda(\mathcal{S}_{nt}^{sing}(\mu))>0$.