Four hyperplanes do not always equipartition a mass in ${\mathbb{R}}^4$

We construct a smooth strictly positive density in ${\mathbb{R}}^4$ that cannot be divided into $16$ parts of the same size by four affine hyperplanes. This settles the last open case of Grünbaum’s 1960 conjecture and disproves Ramos’ general conjecture on hyperplane equipartitions. The proof reduces the problem to proving that five explicit polynomials in six variables have no common zero in $[-1,1{]}^6$. The verification relies on a computer-assisted subdivision argument using convexity and Bernstein coefficients.