Stecher's cul-de-sac conjecture for rectangular-lattice HP folding

Let $J_{\mathrm{rect}}(x)$ denote the maximum number of points achievable for a binary word $x$ in the two-dimensional rectangular lattice. For binary words $x$ and $y$, Stecher's cul-de-sac conjecture.

$$J_{\mathrm{rect}}(x)\leq J_{\mathrm{rect}}(xy).$$

This formalizes the proposed weak monotonicity of the HP folding problem under adjoining a suffix; the paper proves nonmonotonicity under concatenation, so the conjecture is refuted.