The planar attacking cop number conjecture

A planar graph is a graph admitting a planar embedding. For a graph $G$, let $\operatorname{c}(G)$ denote its cop number and let $\operatorname{cc}(G)$ denote its attacking cop number.

Planar attacking cop number conjecture. For every planar graph $G$,

$$\operatorname{cc}(G) \leq \operatorname{c}(G)+1.$$

The conjecture would imply that all planar graphs have attacking cop number at most $4$ once combined with the paper's established bounds. The source gives no resolution of this conjecture.