Flip-flop conjecture for long-term averages of the stochastic logistic map

Let $\rho$ be a positive integer, let $\bar{\lambda}$ be the average parameter value, and let $p_i(\bar{\lambda})$ for $1\leq i\leq 2^\rho$ denote the stable period-$2^\rho$ points of the deterministic logistic map with parameter $\bar{\lambda}$. Let $\mu^*$ be the unique stable invariant distribution of the stochastic logistic map, and write $E_{\mu^*}(X)$ for the expected value of $X$ under this distribution. Flip-flop conjecture. If $\rho$ is odd, then, in regions with period $2^\rho$,

$$E_{\mu^*}(X) > \frac{1}{2^\rho}\sum_{i=1}^{2^\rho}p_i(\bar{\lambda}).$$

If $\rho$ is even, then, in regions with period $2^\rho$,

$$E_{\mu^*}(X) < \frac{1}{2^\rho}\sum_{i=1}^{2^\rho}p_i(\bar{\lambda}).$$

The conjecture predicts that the direction of the comparison alternates as the period increases, at least through the period-doubling range of the parameter. The proposed renormalization perspective relates this alternation to the switching convexity of iterates at the fixed point; the claim is supported empirically beyond the period-two case but is not proved in general.