The connectivity dichotomy for matroids without a uniform minor

Let $k,l\in\mathbb{N}$. A matroid is vertically $l$-connected or cyclically $l$-connected according to the corresponding connectivity conditions, and $U_{k,2k}$ denotes the uniform matroid of rank $k$ on $2k$ elements.

Connectivity dichotomy conjecture. Let $k,l\in\mathbb{N}$. Asymptotically almost all matroids without a $U_{k,2k}$-minor are either vertically $l$-connected or cyclically $l$-connected.

The statement is proposed as a refinement of the fact that almost all matroids are $k$-connected. It appears in an ignored section and no resolution is supplied in the given text.