Simplicity of zeros of Dirichlet L-functions

Let $L(s,\chi)$ range over Dirichlet $L$-functions, with zeros counted according to multiplicity. The simplicity conjecture. Every zero of every Dirichlet $L$-function is simple. The source explains that this follows from interpreting the linear-independence conjecture for the multiset of zero ordinates, and reports numerical evidence and partial theorems proving positive proportions of simple zeros; the universal statement remains open.