Optimality conjecture for stabilized Pell ellipsoid embeddings

Let $P_n$ and $H_n$ be the Pell numbers and half companion Pell numbers defined by

$$P_0=0, P_1=1, P_n=2P_{n-1}+P_{n-2},$$

and

$$H_0=1, H_1=1, H_n=2H_{n-1}+H_{n-2}.$$

Define

$$\beta_n=\begin{cases}\frac{H_{n+2}}{H_n}&\text{if $n$ is even},\\\\[4pt]\frac{P_{n+2}}{P_n}&\text{if $n$ is odd}. \end{cases}$$

Let $E(1,a_2,\dots,a_n)$ denote the symplectic ellipsoid and let $g(a_2,\dots,a_n)$ be the infimum of $R$ for embeddings into $P(R,R)\times\mathbb{C}^{n-2}$. Suppose $a_3,\dots,a_n\ge a_2=\beta_n$. Optimality conjecture. The product embedding

$$E(1,\beta_n,a_3,\dots,a_n)\hookrightarrow P\left(\frac{2\beta_n}{\beta_n+1},\frac{2\beta_n}{\beta_n+1}\right)\times\mathbb{C}^{n-2}$$

is optimal, that is,

$$g(\beta_n,a_3,\dots,a_n)=c_P(\beta_n).$$

These embeddings are known to be optimal for $n=0,1$; the conjecture asks whether this remains true for every $n$.