A Borsuk--Ulam theorem for well separated maps

Suppose that $f_1,\dots ,f_m:S(V)\to R$ are $m$ ($\ge 1$) continuous functions defined on the unit sphere in a Euclidean vector space $V$ of dimension $m+1$ satisfying $f_i(-v)=-f_i(v)$ for all $v\in S(V)$. The classical Borsuk-Ulam theorem asserts that the image of the map $(f_1,\dots ,f_m):S(V)\to R^m$ contains $0=(0,\dots ,0)$. Pursuing ideas in papers of Bárány, Hubard and Jéronimo (2008) and Frick and Wellner (2023), we show that a certain separation property will guarantee that the image contains an $m$-cube.