Generation conjecture for the two-segment piecewise-linear path ideal

Let $T(\mathbb R^d)$ be the tensor algebra equipped with the left and right halfshuffle products, and let $\mathcal{I}(\mathbf{PL}^d_{\leq 2})$ denote the ideal associated with piecewise-linear paths having at most two segments. The level of an element is denoted by $k$. Generation conjecture. $\mathcal{I}(\mathbf{PL}^d_{\leq 2})$ as an ideal with respect to left and right halfshuffle products is generated in levels $k=3,4$. The claim concerns the apparently simple structure of the ideal for two-segment paths and is presented as a conjectural consequence of computations; no resolution is supplied in the source.