2 problems

The nesting conjecture. The variety $V_{\underline{\boldsymbol{h}}}$ is a generically reduced elementary component if and only if

Nonnegative-coefficient conjecture. For every $m\geq 1$, $\mathcal{N}^{(m)}_n(q,t)$ is a polynomial in $q$ and $t$ with nonnegative integer coefficients. The authors further conjec…