Necessary and sufficient condition for convergence of the Magnus series

Let $Y(t;\kappa)$ denote the solution of

$$Y' = \kappa A(t)Y,\qquad Y(0)=I.$$

Denote the eigenvalues of $Y(t;\kappa)$ by $\lambda_n(t;\kappa)$, ordered so that each $\lambda_n$ is continuous as a function of $t$. Let $t_*$ be the smallest $t>0$ for which there exists a $\kappa\in\mathbb{C}$ with $|\kappa|=1$ such that $Y(t;\kappa)$ has a multiple eigenvalue $\lambda_i(t,\kappa)=\lambda_j(t,\kappa)$ with $i\ne j$, whose geometric multiplicity is smaller than its algebraic multiplicity, and such that the loop

$$\{\lambda_i(\tau,\kappa)\mid \tau\in[0,t]\}\cup\{\lambda_j(\tau,\kappa)\mid \tau\in[0,t]\}$$

encircles the origin. Magnus-series convergence conjecture. The Magnus series converges if and only if $t<t_*$. The conjecture proposes a necessary and sufficient criterion: divergence is associated with an encircling collision at a defective multiple eigenvalue, although the paper notes that not every eigenvalue collision causes divergence.