Existence of asymptotically many tight simplices in quaternionic projective space

Let $d$ and $N$ be positive integers, and let ${\mathbb H}{\mathbb P}^{d-1}$ denote quaternionic projective space. A tight $N$-point simplex is a tight code consisting of $N$ points.

Existence conjecture. As $d\to\infty$, there exist tight $N$-point simplices in ${\mathbb H}{\mathbb P}^{d-1}$ for every $N$ satisfying

$$(4-2\sqrt{2}+o(1))d\leq N\leq(4+2\sqrt{2}-o(1))d.$$

The range is suggested by the nonnegativity of the expected local moduli-space dimension. The paper proves several individual existence results, but this asymptotic existence assertion remains open.