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Equality of Cheeger and Dirichlet Cheeger constants on trees

Prove or refute that for every tree graph and all $k$, the multi‑way Cheeger constants equal the $k$‑way Dirichlet Cheeger constants, i.e., establish $h_k(G)=H_k(G)$ for all $k$.

References

Let $G$ be a tree graph, is it true or false that the $k$-way Cheeger constants introduced in \Cref{DEf:Cheeger_constant} are equal to the $k$-way Dirichlet Cheeger constants introduced in \Cref{DEF:dirichlet_cheeger_constants}, i.e. \begin{equation}\label{eq:conj-equal} h_k(G)= H_k(G),\;\; \forall k? \end{equation}

eq:conj-equal:

hk(G)=Hk(G),    ∀k?h_k(G)= H_k(G),\;\; \forall k?

— Nonlinear spectral graph theory (Deidda et al., 4 Apr 2025) in Section 6.2, Relations with Dirichlet Cheeger constants