Gurevich–Howe exhaustion conjecture for low U-rank representations of GL(n)

Let $GL_n$ be the general linear group over a finite field, let $U$-rank be the rank invariant defined in the source, and let ${\Greekmath 0111}$ denote the eta correspondence from the preceding discussion. For an irreducible representation, twisting means tensoring by a one-dimensional character of $GL_n$.

Gurevich–Howe exhaustion conjecture. Suppose $k<\left\lfloor\frac{n}{2}\right\rfloor$. Then, up to twist by a character, every irreducible representation of $U$-rank $k$ of $GL_n$ is in the image of the eta correspondence ${\Greekmath 0111}$.

This conjecture asserts that the eta correspondence, together with character twists, exhausts the irreducible representations of sufficiently low U-rank. The supplied text gives no evidence that it has been resolved.