Modularity conjecture for uniacute spherical codes

Let $\alpha,\beta \in (0,1)$ and let $L \subseteq [-1,-\beta]\cup\{\alpha\}$ with $\alpha \in L$. An $L$-code is a finite set of unit vectors in Euclidean space whose pairwise inner products lie in $L$; write $N_L(d)$ for the maximum size of an $L$-code in $\mathbb{R}^d$. A modular $L$-code is one for which, after possibly reordering its vectors, its Gram matrix $M_C$ dominates a template in the positive-semidefinite order, and write $N_L^{\mathrm{mod}}(d)$ for the maximum size of a modular $L$-code in $\mathbb{R}^d$. Modularity conjecture. For fixed $\alpha$, $\beta$, and $L$ as above,

$$N_L(d)=N_L^{\mathrm{mod}}(d)+o(d)\quad\text{as }d\to\infty.$$

The conjecture asserts that modular codes capture the asymptotically maximal size of uniacute spherical codes, up to a sublinear error. The paper notes that all known tight uniacute spherical codes in high dimensions arise from modular codes, but no resolution is given here.