The conjecture for sums of powers of quadratic polynomials

Let $n$ be a positive integer, and let $\mathcal{QP}_n$ be the set of all nonidentically vanishing rational functions

$$f(x)=\sum_{i=1}^{n}c_{i}P_i^{\alpha}(x),$$

where $c_i\in\mathbb{R}$, $\alpha\leq -1$ is real, and $P_1,\ldots,P_n$ are monic quadratic polynomials with real coefficients and no real zeros. Let $\mathcal{G}_f$ denote the garden of the real rational function $f$.

Conjecture for sums of powers of quadratic polynomials. If $f\in\mathcal{QP}_n$, then $f$ has at most $2n-1$ real critical points. Moreover, if $\alpha$ is a negative integer, each chord of $\mathcal{G}_f$ contains at least one nonreal zero of $\prod_{i=1}^{n}P_i(x)$.

This is proposed as an analogue of the Hawaiian conjecture for the class $\mathcal{QP}_n$, based on numerical experiments. The source does not state whether either assertion has been proved or disproved.