Chappell–Pelsmajer conjecture for degrees 2 through 6

Determine whether every planar graph \(G\) on \(n\) vertices satisfies \(f_d(G)>\frac{2dn}{4d+1}\) for each integer \(d\) with \(2\leq d\leq 6\), where \(f_d(G)\) is the maximum order of an induced forest of maximum degree at most \(d\).

Background

The Chappell–Pelsmajer conjecture proposes that every planar graph on $n$ vertices has an induced forest of maximum degree at most $d$ and order greater than $2dn/(4d+1)$. The paper constructs counterexamples for every integer $d\geq 7$, but explicitly states that the conjecture remains unresolved for $2\leq d\leq 6$.