Grünbaum's orthogonal line equipartition conjecture

Let $P$ be a convex body in the plane with area $1$.

Grünbaum's conjecture. For every $t\in[0,1/4]$, there exists a pair of orthogonal lines that partitions $P$ into four pieces whose areas, in clockwise order, are $t,t,(1/2-t),(1/2-t)$.

The conjecture is open in general, although the supplied context says it has been verified in certain special cases.