4 problems

Let $B=\operatorname{diag}(b_1,\ldots,b_k)$ with $b_i<1$ for all $i$, let $H=\operatorname{diag}(h_1,\ldots,h_n)$, and let $\Gamma\in\mathbb{R}^{n\times k}$ and…

Fortuna's conjecture. The solution $X$ is entrywise nonnegative.

Factorization conjecture. The determinant of $A(\Sigma)_{\cdot,E}$ factorizes as

Lyapunov representation conjecture. There exists a symmetric matrix $P=P^{\top}$ satisfying