Telek's conjecture on Markovian representations of rational arrival processes

Let a rational arrival process (RAP) have joint densities on $(0,\infty)^k$ for every $k\ge 1$, and let $\bm{G}_0$ be the matrix in a RAP representation governing the evolution between arrivals. Assume that all these joint densities are strictly positive and that $\bm{G}_0$ has a unique real eigenvalue of maximal real part. Telek's conjecture. Every such RAP has a finite-dimensional Markovian arrival process (MAP) representation. The conjecture is the point-process analogue of the classical ME/PH characterisation, but the paper shows that it is false: positivity of all joint densities and the dominant-eigenvalue condition do not force a finite-dimensional MAP representation.