Hadamard-product fibre bounds for uniform matroids

For positive integers $k_1,k_2$, set $n=k_1+k_2-1$. Let $h_{k_1,k_2}(p)$ denote the number of pairs of matroids of ranks $k_1$ and $k_2$ on the ground set $[n]$, counted up to permutations, whose flip product is $p$; let $U_{n,k}$ denote the uniform matroid of rank $k$ on $[n]$, and let $*$ denote the Hadamard product. Hadamard-product fibre conjecture. For every such pair $k_1,k_2$, one has

$$h_{k_1,k_2}(p)\geq 1$$

for all $0\leq p\leq U_{n,k_1}*U_{n,k_2}$ when $n\geq 2$, and, when $n\geq 3$,

$$h_{k_1,k_2}(p)\leq n!$$

if and only if $p=U_{n,k_1}*U_{n,k_2}$. The conjecture concerns the possible fibre sizes of the flip-product map; the reverse implication in the second assertion follows from an nbc-basis count, and the first assertion has been verified computationally for the listed small cases, but the general statements remain open.