Brahmagupta-move staircase conjecture for the polydisk

Let $S:\mathbb{R}\to\mathbb{R}$ be the Brahmagupta transformation

$$S(z)=6-\frac{1}{z},$$

and let $\mathrm{accum}$ be the accumulation-point function for polydisk embedding staircases. Define

$$\beta_i:=\mathrm{accum}^{-1}\circ S^i\circ\mathrm{accum}\left(\frac{6+5\sqrt{30}}{12}\right)$$

for each positive integer $i$.

Brahmagupta-move staircase conjecture. The functions $c_{\beta_i}$ have infinite staircases.

Earlier work proves that analogous Brahmagupta images preserve infinite staircases in related ellipsoid embedding functions, but the source states that preservation for the polydisk functions in this family is expected rather than proved.