Conjecture on the type of minimum-weight non-codeword minimal pseudo-codewords

Let $q$ be even, let $\mathbf{H}=\mathbf{H}_{\operatorname{PG}(2,q)}$, and let the type of a pseudo-codeword be $\boldsymbol{t}=(t_0,t_1,t_2,\ldots)$, where $t_\ell$ counts components of value $\ell$. Consider minimal pseudo-codewords that are not multiples of minimal codewords and have minimal AWGNC pseudo-weight among all such minimal pseudo-codewords.

Type conjecture. The type $\boldsymbol{t}$ 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 smallest $\tilde\ell$ should instead satisfy $t_\ell\geq0$ for $\ell\in\{0,1,\ldots,\tilde\ell\}$ and $t_\ell=0$ otherwise.

A positive answer would improve understanding of minimal pseudo-codewords and, in particular, the AWGNC pseudo-weight spectrum gap. The supplied text gives no resolution of this conjecture.