Strong Barratt exponent conjecture for double loop maps

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 $[\Sigma^2X,Y]$.

Strong Barratt exponent conjecture. The double loop map

$$\Omega^2f\colon\Omega^2\Sigma^2X\longrightarrow\Omega^2Y$$

has order at most $p^{r+1}$ in $[\Omega^2\Sigma^2X,\Omega^2Y]$. This is the stronger multiplicative-exponent form of Barratt's conjecture: it concerns the order of the induced map between double loop spaces, rather than only the image on homotopy groups. The source presents it as a conjectural strengthening, and no resolution is supplied in the excerpt.