Meylender–Thalmaier optimal-cover conjecture for truncated squares

Let $Q$ be a pentagon obtained from a square by truncating one of its corners, and let $K_0$ be the trapezoid determined by the placement of the four $Q$-normal triangles described in the source. For a convex shape $K$, define the systolic ratio by

$$\frac{c(K\times Q)^2}{2\operatorname{area}(K)\operatorname{area}(Q)}.$$

Meylender–Thalmaier conjecture. The systolic ratio is strictly less than $1$ for every convex shape $K$, and it is maximized with respect to $K$ if and only if $K$ is a translate of $K_0$ or $-K_0$. This conjecture is based on computational experiments for pentagons obtained by truncating a corner of a square. The source presents it as an unresolved conjecture about the optimal shape of $Q$-covers.