Conjecture on uncommon (2 × k)-systems with maximal s(L)

Let $q$ be an odd prime power. A $(2\times k)$-system $L$ is a system of two linear equations in $k$ variables, and $s(L)$ denotes the parameter used in the paper to measure the system's complexity. A system is uncommon if the density of monochromatic solutions in two-colourings is not asymptotically minimized by the expected density from a random two-colouring.

Uncommon $(2\times k)$-system conjecture. For even $k\geq 6$ and sufficiently large odd $q$, every $(2\times k)$-system $L$ satisfying

$$s(L)=k-1$$

is uncommon.

The paper proves the corresponding result for $(2\times4)$-systems and notes that the analogous assertion for larger $k$ is unknown. This conjecture proposes the extension to all even $k\geq6$.