The openness and smooth-witness conjecture for repeated integrals

For a nonnegative integer $n$, let

$$W_n=\left\{b\in\mathbb{R}^{n+1}:\text{ there is an }f\in C^n[0,1]\text{ with }D^nf\text{ increasing and nonconstant, }D^jf(0)=0\text{ and }D^jf(1)=b_j\text{ for }j=0,\dots,n\right\}.$$

Here increasing means non-strictly increasing. The statement $(P_n)$ is that $W_n$ is open in $\mathbb{R}^{n+1}$ and that every $b\in W_n$ has a witnessing function $f$ which is $C^\infty$, satisfies $D^{n+1}f(x)>0$ for all $x\in(0,1)$, and obeys

$$D^{n+1}f(0)=D^{n+1}f(1)=1,\qquad D^jf(0)=D^jf(1)=0\quad\text{for }j>n+1.$$

The $(P_n)$ conjecture. $(P_n)$ is true for all nonnegative integers $n$.

This assertion is the assumption used in the paper's approximation theorem for functions whose derivatives are piecewise monotone. The supplied text gives no resolution of $(P_n)$, so its status remains open.