Non-injectivity conjecture for average-distance minimizers below the critical exponent

Let $\mu$ be a measure and, for $\lambda>0$ and $1\leq p<2$, consider the average-distance minimization problem whose global minimizers are parameterized curves. Non-injectivity conjecture. For every $1\leq p<2$, there exist $\lambda>0$ and a measure $\mu$ for which the global minimizer is not injective. This conjecture asserts that the exponent range in the preceding injectivity theorem is sharp; the source provides no resolution.