Let ${\mathcal K}$ be the space of Fredholm operators on a Hilbert space. It is well known that ${\mathcal K}$ represents $K$-theory. Let ${\mathcal K}_0$ be the path component of ${\mathcal K}$ of operators of index zero. Then ${\mathcal K}_0$ is a model for the space that algebraic topologists usually call $BU$ - the classifying space of the infinite unitary group.

My question is: is it possible to realise the filtration of $BU$ by the subspaces $BU(n)$ in terms of Fredholm operators?

An obvious idea is to consider the subspace of ${\mathcal K}_0$ consisting of operators whose kernel/cokernel has rank at most $n$. Is this space weakly homotopy equivalent to $BU(n)$?

Hopefully this question is interesting enough on its own (I am prepared to find out that it is stupid). My motivation is to understand better Jesse McKeown's answer to my previous question. This is my attempt to understand the statement that the space of subspaces of a Hilbert space of corank at most $n$ is a model for $BU(n)$. If there are other ways to make it precise, I would be very interested in learning about it.

EDIT: I think that Tyler Lawson's negative answer to the previous question makes it very likely that the answer to this question is negative as well.