Is every self-homeomorphism homotopic to a diffeomorphism?

In dimensions 2 and 3 every homeomorphism is isotopic to a diffeomorphism (this should be in Moise's book "Geometric topology in dimensions 2 and 3", it also follows from Kirby and Siebenmann's work). In dimension 4 there are self-homeomorphisms of simply-connected smooth compact manifolds which are not homotopic to diffeomorphisms. This follows e.g. from invariance of the $\pm$ canonical class of smooth algebraic surfaces under diffeomorphisms, while, by Freedman's work, any automorphism of the intersection form is induced by a homeomorphism.

Edit. One more useful thing: The group of homeomorphisms of a topological manifold is locally contractible (with respect to the $C^0$ topology), this is a theorem by Chernavskii (1969). Thus, if you can approximate a homeomorphism by diffeomorphisms, they will be isotopic (for sufficiently close approximation).

First, note that a homeomorphism $f$ of compact smooth manifolds is homotopic to a diffeomorphism if and only if one can approximate $f$ arbitrarily well by diffeomorphisms.

For $n \leq 3$, this paper of Munkres claims as a corollary that a homeomorphism $f: M \to N$ of smooth manifolds may be approximated arbitrarily well by a diffeomorphism. This settles my question here. There is a harder question of whether or not every homeomorphism is isotopic to a diffeomorphism. The consensus on this MathOverflow question and answers seem to be that it's true for $n=2$, though I haven't checked the stated references. Ian Agol's answer there sounds like it's true for $n=3$, but I'm not sure.

This note of Stefan Müller gives a proof that for $n \geq 5$, a homeomorphism of compact $n$-manifolds can be approximated arbitrarily well by diffeomorphisms if and only if the same homeomorphism is isotopic to a diffeomorphism.

Every related question in $n=4$ seems to be wide open.

In many dimensions, exotic spheres are counterexamples.

For example, the group of $7$ spheres is diffeomorphic to $\mathbb{Z}_{28}$. Pick any element $\Sigma$ which doesn't not have order $2$, for example, a generator.

There is an orientation reversing homeomorphism $f:\Sigma\rightarrow \Sigma$ (so $f$ has degree $-1$), but there is no such orientation reversing diffeomorphism.

Since the degree is a homotopy invariant, this provides examples of what you want.

Proof: Since $\Sigma$ is homeomorphic to $S^7$ and $S^7$ admits an orientation reversing homeomorphism, so does $\Sigma$. Now, in the group of exotic spheres, the inverse element is the same sphere with orientation reversed. Hence, if $\Sigma$ admits an orientation reversing diffeomorphism, then $\Sigma = -\Sigma \in \mathbb{Z}_{28}$, that is, the order of $\Sigma$ in $\mathbb{Z}_{28}$ is either $1$ or $2$.

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Of course, this proof carries over to any dimension for which the group of exotic spheres contains an element not of order $2$. This has to occur if the order of the group is not a power of $2$, so, according to the chart on wikipedia, there are examples in dimension $7, 10, 11, 13, 15, 19, 20$. See also the OEIS sequence to find more examples.