Finite-time singularity conjecture for curve diffusion with free boundary

Suppose $|e|>0$. Let $\eta_1,\eta_2:\mathbb{R}\to\mathbb{R}^2$ be parallel straight lines defining the exterior problem, and let $\gamma_0:(-1,1)\to\mathbb{R}^2$ be an immersed curve satisfying its boundary conditions. Let $\gamma:(-1,1)\times[0,T)\to\mathbb{R}^2$ be the curve diffusion flow with free boundary and initial data $\gamma_0$. The finite-time singularity conjecture. The flow exists for at most finite time, and $\gamma(\cdot,t)$ approaches a multiply-covered straight line in the $C^0$ topology but not in $C^k$ for any $k\geq1$. This proposed global-behaviour description concerns the expected singularity mechanism for curve diffusion under the stated exterior boundary conditions; the source gives no resolution.