Structural decomposition conjecture for minimal pseudo-codewords

Let $q$ be an even prime power and let $\mathbf{H}=\mathbf{H}_{\operatorname{PG}(2,q)}$. A component is significant if, for most lines passing through its point, its value is the sum of the other components on such a line. Structural decomposition conjecture. Every minimal pseudo-codeword is a sum of a few minimal pseudo-codewords, followed by changing a few low-value components into significant components. More specifically, one should obtain minimal pseudo-codewords by summing two minimal pseudo-codewords whose associated systems have rank $n-2$ when possible, or lower otherwise, and changing a few non-significant components into significant ones. This is an empirical structural description based on small projective planes; the source gives no proof or resolution.