Reduced local-complete-intersection conjecture for lines on the permanental hypersurface

Let $P_{n,n}^n$ be the permanental hypersurface of $n\times n$ matrices, and let $\mathbf{F}_1(P_{n,n}^n)$ be its Fano scheme of lines. Reduced local-complete-intersection conjecture. The scheme $\mathbf{F}_1(P_{n,n}^n)$ has dimension

$$2n^2-n-5,$$

which is the expected dimension, and is a reduced local complete intersection. The claim concerns the all-$n$ case; the paper reports verification of the corresponding expected-dimension behavior for $n=3$ and $4$.