Fixed-parameter tractability of Map-$\varepsilon$-PDD for $k+\delta+h$

Let $k$, $\delta$, and $h$ be the parameters of the Map-$\varepsilon$-PDD problem, where $h$ is the value of $H$ in a phylogenetic tree. Map-$\varepsilon$-PDD conjecture. Map-$\varepsilon$-PDD is fixed-parameter tractable when parameterized by $k+\delta+h$. The conjecture is motivated by the known fixed-parameter tractability of $\varepsilon$-PDD with respect to $k+h$ and by the expectation that the corresponding hardness result for Map-1-PDD does not extend to Map-$\varepsilon$-PDD; the proof of the analogous result in the cited work contains an incorrect lemma, so the claim remains open.