Conjecture on the location of zeros of the Padé denominator

Consider the $[P/P]$-Padé approximant $R_P(z)=T_P(z)/Q_P(z)$, where $T_P$ and $Q_P$ have degree $P$ and are computed from the first $2P+1$ Taylor coefficients of $f$. Let $t$ denote the parameter on which the approximant depends, and let $z_j$ be a zero of $Q_P$. Zero-location conjecture. For all values of $t$ and $P$, every zero $z_j$ of $Q_P$ lies below the real axis:

$$Q_P(z_j)=0\quad\Longrightarrow\quad \operatorname{Im}(z_j)<0.$$

This conjecture would locate all poles of the Padé approximant in the lower half-plane, a property relevant to the stability of the spectral split-step Padé method. The supplied text does not establish the claim or indicate whether it has been resolved.