XXY equivalence conjecture for maximal equiangular tight frames

Let $d>3$. An equiangular tight frame (ETF) with parameters $(r,n)$ is a set of $n$ unit vectors in $\mathbb{S}^{r-1}$ whose coherence attains the Welch bound. XXY's ETF existence conjecture. The following existence statements are equivalent:

$$\text{(i) An ETF with parameters }\left(d,\frac{d(d+1)}{2}\right).$$ $$\text{(ii) An ETF with parameters }\left(d-1,\frac{(d-1)(d+1)}{3}\right).$$

This conjecture relates existence of maximal ETFs to ETFs in one lower dimension. Maximal ETFs are closely connected with tight spherical $5$-designs, and the equivalence remains unresolved in general.