Conjecture on extremal real equiangular tight frames

Let $d\geq 5$. An equiangular tight frame (ETF) with parameters $(d,n)$ is a real ETF for $\mathbb{R}^d$ of size $n$. ETF extension conjecture. There exists an ETF with parameters

$$\left(d+1,\frac{(d+1)(d+2)}{2}\right)$$

if and only if there exists an ETF with parameters

$$\left(d,\frac{d(d+2)}{3}\right).$$

The conjecture relates two extremal ETF sizes arising from antipodal spherical designs and would connect the existence of the known configurations in dimensions $6$ and $22$ with corresponding larger ETFs. Its status is not resolved in the source.