Baryshnikov–Ghrist monotonicity conjecture for unimodal p-category

Let $X$ be a topological space and let $f:X\to[0,\infty)$ be a continuous function. For $p\in(0,\infty]$, let $\operatorname{{\bf ucat}}^p(f)$ denote the minimum number of unimodal functions needed to represent $f$ by an $\ell^p$-combination, with $p=\infty$ interpreted as pointwise maxima. Baryshnikov–Ghrist monotonicity conjecture. If $0<p_1<p_2\leq\infty$, then

$$\operatorname{{\bf ucat}}^{p_1}(f)\leq\operatorname{{\bf ucat}}^{p_2}(f).$$

Equivalently, the unimodal $p$-category is monotone nondecreasing in $p$. The conjecture is motivated by the role of unimodal category in obtaining bounds in higher dimensions; this paper proves it for functions on $\mathbb R$ and $S^1$, but gives counterexamples on certain graphs and on the Euclidean plane, so the general statement is refuted.