Kano's orthogonal-curve mass bisection conjecture

Let $m$ be a positive integer, and let $\mu_1,\dots,\mu_m$ be absolutely continuous measures in $\mathds{R}^2$.

Kano's conjecture. There exists a curve in $\mathds{R}^2$ formed by horizontal and vertical segments, with at most $m-1$ turns, that bisects every measure.

This is an orthogonal-curve analogue of simultaneous bisection by a polynomial graph. The source attributes the conjecture to Mikio Kano and explicitly states that it remains open.