The matroid strong-map spectral recursion conjecture

Let $N$ and $N'$ be matroids, and let $N \rightarrow N'$ be a strong map. Write $IN(N)$ and $IN(N')$ for their independence complexes, and let $(IN(N),IN(N'))$ denote the resulting interval. Strong-map spectral recursion conjecture. If there is a strong map

$$N \rightarrow N'$$

then the interval $(IN(N),IN(N'))$ has integral Laplacian eigenvalues and satisfies the spectral recursion.

The conjecture extends the proved rank-difference-one case for matroid pairs. It is motivated by experimental evidence from randomly chosen matroids, but the paper reports no proof when the rank difference exceeds one.