There is no analogue to Jarn\'ik's relation for twisted Diophantine approximmation

Jarn\'ik's relation is in dimension $2$ the formula $\hat{\lambda} + \frac{1}{\hat{\omega}} = 1$ linking both uniform exponents. It is an open question to generalize this equation to higher dimension, or to the multiplicative case. In this paper we consider a twisted case, between the classical and the multiplicative one, and we show that no analogue to Jarn\'ik's relation holds.