Transcendence conjecture for the real stabilization threshold of elliptic spaces

Let $X$ be a simply connected elliptic space, and let $\frak{pp}_{\mathbb R}(X;1)$ denote its real stabilization threshold. Let $P(s)=P_X^{\pi}(s)$ and $Q(s)=P_X(s)$. Transcendence conjecture for the real stabilization threshold. If $X$ is not homotopic to $S^{2n+1}$, then

$$\frak{pp}_{\mathbb R}(X;1)$$

is a transcendental number. In particular, if $\frak{pp}_{\mathbb R}(X;1)$ is attained at a critical point $s$ of the implicit function $r(s)$, then

$$\frac{P'(s)Q(s)}{P(s)Q'(s)}$$

is a transcendental number. This conjecture proposes a strong arithmetic property for stabilization thresholds; the source does not provide evidence of a resolution.