Ergodic $\times p$-invariant measures on ${\mathbb{T}}^2$ with no dimension dropping projections

Fix an integer $p\ge 2$ and $0<s<1$. We construct an ergodic $\times p$-invariant measure $\mu$ on ${\mathbb{T}}^2$ of dimension $s$, such that every line projection preserves dimension, including when the corresponding projected IFS has exact overlaps. In fact, our measure assigns mass $O(w^s)$ to every planar tube of width $w$. The result is sharp at the endpoint $s=1$ as shown by Pyörälä, Shmerkin, Suomala and Wu (2025). In contrast, we show that every quasi-Bernoulli $T_p$-invariant measure with positive dimension admits a dimension-dropping projection.