The Eulerianity conjecture for Peano continua

Let $X$ be a Peano continuum. A free arc is an inclusion-maximal open subset of $X$ homeomorphic to $(0,1)$; write $E(X)$ for the collection of free arcs, called edges. An edge cut is a non-empty set $F\subseteq E(X)$ of edges crossing a partition of the ground-space of $X$ into two disjoint clopen subsets.

The Eulerianity conjecture. A Peano continuum $X$ is Eulerian if and only if every edge cut of $X$ is even.

This conjecture extends Nash-Williams's edge-cut characterisation of finite Eulerian graphs to Peano continua. The paper identifies it as the main open problem and proves it for several classes, but leaves it open for general Peano continua.