The $(Q_n)$ conjecture on comonotone jet interpolation

For real numbers $c<d$ and $s\in\{1,-1\}$, let $V_n^s[c,d]$ be the set of endpoint jets arising from functions in $\mathscr{F}_n^s[c,d]$, and let $V_n^{s,\infty}[c,d]$ be the corresponding set allowing all derivative orders. The statement $(Q_n)$ is that, for every $n$, these sets agree and $V_n^s[c,d]$ is open in $\mathbb{R}^{2(n+1)}$.

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

This asserts both finite-order comonotone interpolation and the openness of the resulting endpoint-jet cone. The supplied text does not indicate whether the statement has been proved or disproved.