The dominant maximal weight count conjecture for affine type B

Let $\mathfrak g=B_n^{(1)}$, let $\ell\geq 2$, and let ${\rm \max}^+(\lambda+\Lambda\mid\ell)$ denote the set of dominant maximal weights at the indicated level. Dominant maximal weight count conjecture.

The number of elements in ${\rm \max}^+((\ell-2)\mathbf{\Lambda}_0+\Lambda\mid\ell)$ is

$$\binom{n+\lfloor \ell/2\rfloor}{\lfloor \ell/2\rfloor}+\binom{n+\lfloor (\ell-1)/2\rfloor}{\lfloor (\ell-1)/2\rfloor},$$

and the number of elements in ${\rm \max}^+((\ell-2)\mathbf{\Lambda}_n+\Lambda\mid\ell)$ is

$$\binom{n+\lfloor \ell/2\rfloor}{\lfloor \ell/2\rfloor}+\binom{n+\lfloor \ell/2\rfloor-1}{\lfloor \ell/2\rfloor-1}.$$

This conjecture predicts explicit binomial formulas for the numbers of dominant maximal weights in the two specified affine-weight families. The supplied text gives no resolution, so the conjecture is recorded as open.