Turán bound for suspensions of arbitrary trees

Let $T$ be a tree whose smaller color class has size $k$, and let $\widehat{T}$ denote the suspension of $T$. Let $f(n,k)$ be the bound defined earlier in the paper. Conjecture on suspensions of trees. For large $n$,

$${\rm ex}(n,\widehat{T})\le f(n,k).$$

The paper proves the result for balanced trees and notes that the same conclusion would follow for certain unbalanced trees if the relevant decomposition lemma holds. The conjecture asks whether the bound extends to every tree, and the source gives no resolution.