Small-type minimal pseudo-codeword conjecture for projective-plane Tanner graphs

Let $\mathbf{H}=\mathbf{H}_{\operatorname{PG}(2,q)}$. Consider minimal pseudo-codewords having minimum AWGNC pseudo-weight among all minimal pseudo-codewords that are not multiples of minimal codewords. Small-type conjecture. The type $\mathbf{t}$ of these pseudo-codewords has $t_0$ non-negative, $t_1$ positive, $t_2$ positive, and $t_{\ell}=0$ otherwise. If this is false, the source asks for the smallest $\tilde{\ell}$ such that their type satisfies $t_{\ell}\geqslant 0$ for $\ell\in\{0,1,\ldots,\tilde{\ell}\}$ and $t_{\ell}=0$ otherwise. The conjecture is motivated by observations for small $q$ and would clarify the structure of minimum-weight non-codeword minimal pseudo-codewords and the AWGNC pseudo-weight spectrum gap; the source gives no resolution.