The partial saturation ratio bound for planar graphs with bounded twin multiplicities

Let $k_1,k_2$ be non-negative integers, let $\mathcal{G}_{k_1,k_2}$ be the class of graphs under consideration, and let $G\in\mathcal{G}_{k_1,k_2}$. Write $\operatorname{psr}(G)$ for the partial saturation ratio.

Partial saturation ratio conjecture. Every such graph satisfies

$$\operatorname{psr}(G)>\frac{1}{9+k_1+6k_2}.$$

The paper states that this is the best possible bound, based on a construction, and notes that the case $k_1=k_2=0$ is of greatest interest. The conjecture extends the theorem proved for some pairs of non-negative integers to all such pairs.