Barratt's exponent conjecture for double suspensions

Let $p$ be a prime, let $X$ be a pointed space, and let $Y$ be a space. Suppose that $f\colon \Sigma^2X\longrightarrow Y$ has order $p^r$ in $[\Sigma^2X,Y]$.

Barratt's exponent conjecture. The image of the induced map

$$f_*\colon\pi_*(\Sigma^2X)\longrightarrow\pi_*(Y)$$

should be annihilated by $p^{r+1}$; equivalently,

$$p^{r+1}\operatorname{Im}(f_*)=0.$$

This conjecture predicts that the order of a map from a double suspension controls the torsion in the homotopy groups detected by that map. In particular, when the identity of $\Sigma^2X$ has order $p^r$, it implies that the mod-$p$ homotopy exponent of $\Sigma^2X$ is $p^{r+1}$.