Existence and uniqueness of s-curve segments of tensioned elastica satisfying geometric Hermite interpolation conditions

It has been recently proved that every proper restricted elastic spline is a stable nonlinear spline, and this yields a broad existence proof for stable nonlinear splines. When tension is included in the setup, stable nonlinear splines under tension always exist, but they do not always have the property that each piece (connecting one interpolation point to the next) is an s-curve. Being correlated with the fairness of an interpolating curve, this property is desirable and we conjecture that the framework employed successfully with restricted elastic splines will also work well with nonlinear splines under tension. Our purpose is to prove the following foundational result: Given points $P_1\ne P_2$, in the plane, along with corresponding unit directions $d_1,d_2$ that satisfy $d_1\cdot (P_2-P_1)\ge 0$ and $d_2\cdot (P_2-P_1)\ge 0$, there exists a unique s-curve segment of Euler–Bernoulli elastica under tension $\lambda >0$ that connects $P_1$ to $P_2$ with initial direction $d_1$ and terminal direction $d_2$.