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Let $\Lambda\subset\mathbb{R}^2$ be a non-closed piecewise-$C^1$ curve, whichis either bounded with two free endpoints or unbounded with one free endpoint.Let $u_\pm|_\Lambda \in L^2(\Lambda)$ be the traces of a function $u$ in theSobolev space $H^1({\mathbb R}^2\setminus \Lambda)$ onto two faces of$\Lambda$. We prove that for a wide class of shapes of $\Lambda$ theSchr\"odinger operator $\mathsf{H}_\omega^\Lambda$ with$\delta^\prime$-interaction supported on $\Lambda$ of strength $\omega \inL^\infty(\Lambda;\mathbb{R})$ associated with the quadratic form \[H^1(\mathbb{R}^2\setminus\Lambda)\ni u \mapsto \int_{\mathbb{R}^2}\big|\nabla u\big|^2 \mathsf{d} x  - \int_\Lambda \omega \big| u_+|_\Lambda - u_-|_\Lambda \big|^2 \mathsf{d} s\] has no negative spectrum provided that $\omega$ is pointwise majorized by astrictly positive function explicitly expressed in terms of $\Lambda$. If,additionally, the domain $\mathbb{R}^2\setminus\Lambda$ is quasi-conical, weshow that $\sigma(\mathsf{H}_\omega^\Lambda) = [0,+\infty)$. For a boundedcurve $\Lambda$ in our class and non-varying interaction strength$\omega\in\mathbb{R}$ we derive existence of a constant $\omega_* > 0$ suchthat $\sigma(\mathsf{H}_\omega^\Lambda) = [0,+\infty)$ for all $\omega \in(-\infty, \omega_*]$; informally speaking, bound states are absent in the weakcoupling regime.

On absence of bound states for weakly attractive $\delta^\prime$-interactions supported on non-closed curves in $\mathbb{R}^2$