Random graph-state overlap conjecture for Pauli-diagonal product states

Let $G$ be sampled uniformly from graph states, and let $d4af$ be the set of eigenstates of the operators $X+Y$ and $X-Y$. For an $n$-qubit product state, write $d4af^{d4af\textsuperscript{⊗}n}$ for the set of tensor products of $n$ states from $d4af$. Random graph-state overlap conjecture. There exist constants $c,C>0$ such that

$$\mathbb{P}\left[\max_{\lvert\alpha\rangle\ind4af^{\otimes n}}\left\lvert\langle\alpha\mid G\rangle\right\rvert^2\geq 2^{-n+n^c}\right]\geq C.$$

Here the probability is taken uniformly over graph states. The conjecture asserts that the overlap with these highly magical product states is substantially larger than the baseline $2^{-n}$ with constant probability, which would rule out the proposed classical simulation strategy based on replacing $X+Y$ and $X-Y$ measurements by coin flips; the paper states that this remains unproved.