The $p=2$ conjecture on symmetries of $\mathcal{L}$-invariant slopes

Given a positive even integer $k$, let $\nu_\mathcal{L}^{\varepsilon}(k,p,N)$ be the finite sequence of slopes of the $p$-adic $\mathcal{L}$-invariants attached to forms in $\mathcal{S}_k(\Gamma_0(pN))^{p-\mathrm{new},\varepsilon}$, ordered decreasingly, where $\varepsilon\in\{+1,-1\}$ is the eigenvalue of the Atkin–Lehner operator $W_p$. Let $d_{+1}$ and $d_{-1}$ denote the dimensions of the corresponding Atkin–Lehner eigenspaces. The $p=2$ conjecture. For $k\in 2+4\mathbb{Z}$ with $k\geq 10$ and $\varepsilon\in\{\pm1\}$, one has

$$\nu_\mathcal{L}^{\varepsilon}(k,2,1)=\nu_\mathcal{L}^{-\varepsilon}(k+6,2,1).$$

For every even integer $k$, the final $\min\{d_{+1},d_{-1}\}$ slopes in $\nu_\mathcal{L}^{+}(k,2,1)$ and $\nu_\mathcal{L}^{-}(k,2,1)$ agree. The statement is supported computationally by the tables through $k=70$ and extends analogous observations for other levels, but no theoretical interpretation or proof is given.