Structural conjecture for minimal pseudo-codewords of projective-plane Tanner graphs

Let $q$ be even and let $\mathbf{H}=\mathbf{H}_{\operatorname{PG}(2,q)}$ define the Tanner graph. A component is called significant when it equals the sum of the other components on a line through the corresponding point, for most such lines.

Structural conjecture. Every minimal pseudo-codeword is a sum of a few minimal pseudo-codewords, with one or two low-value components changed so that they become the large components in the equations associated with lines passing through them. To find minimal pseudo-codewords, it suffices to take sums of two minimal pseudo-codewords whose associated systems have rank $n-2$, if possible, or lower otherwise, and change one insignificant component into a significant one.

The conjecture is intended to clarify the structure of minimal pseudo-codewords and the AWGNC pseudo-weight spectrum gap. The supplied text gives no resolution, so the claim remains open.