The local model conjecture for homeomorphism groups of non-compact manifolds

Let $M$ be a non-compact $\sigma$-compact $n$-manifold, possibly with boundary. Write $\mathcal H(M)$ for the homeomorphism group of $M$ with the Whitney topology, and $\mathcal H_c(M)$ for the subgroup of compactly supported homeomorphisms. Let $l_2$ be the separable Hilbert space, $\square^\omega l_2$ its countable box power, $\boxdot^\omega l_2$ its countable small box power, and $\mathbb R^\infty$ the direct limit of the sequence $\mathbb R^1\subset\mathbb R^2\subset\cdots$.

Local model conjecture. The pair $(\mathcal H(M),\mathcal H_c(M))$ is locally homeomorphic to $(\square^\omega l_2,\boxdot^\omega l_2)$ at the identity $\mathrm{id}_M$ of $M$. In particular, the group $\mathcal H_c(M)$ is an $(l_2\times\mathbb R^\infty)$-manifold.

This is proposed as a non-compact version of the Homeomorphism Group Problem. Earlier results establish the asserted local model for certain one-dimensional examples and show that $\mathcal H_c(M)$ has the indicated manifold structure for non-compact separable graphs; the general case for non-compact $\sigma$-compact manifolds is left open.