Calegari--Stein opposite Atkin--Lehner sign conjecture

Let $k<p-1$, and let $f$ and $g$ be two newforms of level $\Gamma_0(p)$. Write $\varepsilon(f)$ and $\varepsilon(g)$ for their Atkin--Lehner signs, and say that the forms are congruent modulo $p$ when their Hecke eigensystems are congruent modulo $p$. Calegari--Stein conjecture. If $f$ and $g$ are congruent modulo $p$, then

$$\varepsilon(f)=-\varepsilon(g).$$

The conjecture concerns the relationship between mod-$p$ congruences and Atkin--Lehner signs for weights below $p-1$; the source presents it as an open conjecture compatible with the paper's computations.