The Atkin–Lehner equidistribution conjecture for level-2 modular forms

Let $k\geq 2$, and let $S_k(\Gamma_0(2))$ be the space of cusp forms of weight $k$ and level $2$. Let $W_2$ denote the Atkin–Lehner operator, and set

$$d=\dim_\mathbb{C}S_k(\Gamma_0(2)).$$

Atkin–Lehner equidistribution conjecture. For all $k\geq 2$, the space $S_k(\Gamma_0(2))$ decomposes under $W_2$ into Hecke-irreducible subspaces of dimensions $\lfloor d/2\rfloor$ and $\lceil d/2\rceil$.

This is an analogue of the Maeda conjecture for level $2$, predicting that the Hecke-irreducible pieces in the two Atkin–Lehner eigenspaces are as evenly sized as possible. The source reports computational evidence through weight $400$, while the assertion is not established in general.