5 problems

Extended local-permutation conjecture. Every element of $E(\mathfrak{t}_{\mathrm{loc}})$ is similar to a block-shift matrix via an element of $\Pi_{\mathrm{loc}}^{\mathrm{ex}}$.

Irrational-angle counterexample conjecture. It is impossible to find a counterexample satisfying $\varphi_0\notin 2\pi\mathbb{Q}$.

Factorization conjecture. If $\theta$ does not factor in this way, then $W(S_\theta^1)$ and $W(S_\theta^2)$ are open. If $\theta$ does factor in this way, then $W(S_\theta^1)$ and…

Unital uniform-algebra conjecture. If $\|\theta_\alpha\|\leq 1$, then

Numerical-range conjecture. The numerical range of $C_\varphi$ on $H^2(D)$ is the interior of the convex hull of an algebraic curve of class $p$ and degree $p^2-p$. Moreover, the r…