Uniform upper-bound conjecture for acip densities of piecewise expanding unimodal maps

Let $\mathfrak{X}$ be the class of maps under consideration, and for each $f\in\mathfrak{X}$ let the acip denote its absolutely continuous invariant probability measure, with density relative to Lebesgue measure. Uniform boundedness conjecture. There exists a constant $K$ such that for every $f\in\mathfrak{X}$ the density of the acip for $f$ is bounded above by $K$. This is the weaker version motivated by the family whose maximal density approaches $2$ as the plateau parameter tends to $1/2$; the source presents it together with a stronger conjecture giving the explicit bound $2$.