Crowns are extremal for matroids omitting a complete-graphic minor

Let $ll in N$ with $ll ge 2$, and let $q$ be the largest prime power with $q le ll$. For sufficiently large $t in N$, let $\falM$ be the class of matroids with no $U_{2,ll+2}$- or $M(K_{t+3})$-minor. Crowns' extremal-function conjecture. For all sufficiently large $n$,

h_{\falM}(n)=q^t n+\fleft(\frac{q^t-1}{q-1}-tq^t\right).

The preceding theorem supplies matching bounds up to the stated gap, and the authors explain that the lower bound comes from crowns and believe it is correct for large $t$ and $n$; proving that crowns are extremal remains open.