Maximum-size conjecture for 3-connected positroids excluding a uniform minor

Let $r$ and $\ell$ be integers with $r \ge 2$ and $\ell \ge 3$. A 3-connected rank-$r$ positroid is a positroid of rank $r$ that is 3-connected, and a $U_{2,\ell+2}$-minor-free positroid has no minor isomorphic to the rank-two uniform matroid $U_{2,\ell+2}$.

Maximum-size conjecture. Every 3-connected rank-$r$ positroid with no $U_{2,\ell+2}$-minor has at most

$$r + r\left\lfloor \frac{\ell - 1}{2} \right\rfloor$$

elements if $\ell$ is odd, and at most

$$r + r\left\lfloor \frac{\ell - 1}{2} \right\rfloor + 1$$

elements if $\ell$ is even.

The conjectured bounds are attained by the families $W(r,\ell)$ for odd $\ell$ and $W(r,\ell)^+$ for even $\ell$, which the paper shows are 3-connected positroids with the required excluded-minor property. The conjecture asks whether these constructions maximize the number of elements among all such positroids; its status is not established in the supplied text.