Sign conjecture for Shapiro's generalized recurrence curve

Let $A(z)$ and $B(z)$ be complex polynomials, and let $k$ and $\ell$ be coprime integers with $1<\ell<k$. Consider the real algebraic curve from Shapiro's conjecture, defined by

$$\operatorname{Im}\left(\frac{B^k(z)}{A^\ell(z)}\right)=0.$$

Sign conjecture. On the portion of this curve containing the zeros, the following inequalities hold: if $k$ is even, then

$$0\leq \operatorname{Re}\left(\frac{B^k(z)}{A^\ell(z)}\right)<\infty.$$

If $k$ is odd and $\ell$ is odd, then

$$0\leq-\operatorname{Re}\left(\frac{B^k(z)}{A^\ell(z)}\right)<\infty;$$

and if $k$ is odd and $\ell$ is even, then

$$0\leq \operatorname{Re}\left(\frac{B^k(z)}{A^\ell(z)}\right)<\infty.$$

This conjecture refines the proposed zero-containing curve by specifying the relevant sign of its real parameter. It is introduced on the basis of numerical experiments, and no general proof or disproof is given in the source.