Completeness conjecture at the extremal parameters for planar factor-critical graphs

Let $G$ be a minimal $k$-$\{1,2\}$-factor-critical planar graph, and let $t$ be one of the two values $k$ or $k+1$.

Completeness conjecture. If $G$ is a minimal $t$-$\{1,2\}$-factor-critical graph, then $G$ is a complete graph.

The conjecture is motivated by examples attaining both degree bounds in the source's planar minimum-degree theorem, including complete graphs. Its general status is not resolved in the supplied text.