Simion's unimodality conjecture for lattice-path numbers

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ be an integer partition with $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_r\geq 0$, and let $\lambda'$ be its conjugate. For integers $m\geq\lambda'_1$ and $n\geq\lambda_1$, let $N(m,n,\lambda)$ denote the number of lattice paths in an $m$-by-$n$ rectangular grid from the lower-left to the upper-right corner, using unit up and right steps and avoiding the Ferrers diagram of $\lambda$ removed from the upper-left corner.

Simion's conjecture. For each integer $\ell$ and each partition $\lambda$, the sequence

$$N(\lambda'_1,\lambda_1+\ell,\lambda), N(\lambda'_1+1,\lambda_1+\ell-1,\lambda), \ldots, N(\lambda'_1+\ell,\lambda_1,\lambda)$$

is unimodal.

Simion's conjecture was strengthened by Hildebrand, who proved that these numbers are log-concave; consequently, the stated unimodality conjecture is solved.