The open Eulerianity conjecture for Peano continua

Let $X$ be a Peano continuum, and let $X_\sim$ be the quotient obtained by collapsing each component of the ground-space of $X$ to a point. The resulting space is graph-like; its vertices are the collapsed components. A Peano continuum is open Eulerian if it is the strongly irreducible image of a map from the unit interval.

The open Eulerianity conjecture. A Peano continuum $X$ is open Eulerian if and only if all but two vertices of $X_\sim$ have even degree.

This is presented as an equivalent formulation of the Eulerianity conjecture via the construction that adds an arc between two points. It would provide a characterisation of open Eulerian continua and remains open.