Attained maximum Hardy success probability

Identify false and true with $0$ and $1$, and let $e_0,e_1$ be the standard basis of $\mathbb C^2$. A parameter choice $d$ consists of complex numbers $\alpha_P,\beta_P$ for $P\in\{0,1\}$ satisfying $|\alpha_P|^2+|\beta_P|^2=1$ and $0<|\alpha_P|<1$ for each $P$. For each such choice, define $u_P=\alpha_Pe_0+\beta_Pe_1$, $v_P=\overline{\beta_P}e_0-\overline{\alpha_P}e_1$, and four vectors $f_0=v_0\otimes v_1$, $f_1=e_0\otimes u_1$, $f_2=u_0\otimes e_0$, and $f_3=e_0\otimes e_0$, where $(r\otimes s)_{a,b}=r_as_b$. Define $g_0=f_0$ and, successively for $j=1,2,3$, define

where $\langle x,y\rangle=\sum_{a,b}\overline{x_{a,b}}y_{a,b}$; the real square root is $0$ on nonpositive inputs, and the reciprocal of $0$ is $0$. The assertion is the conjunction of the following three claims: every parameter choice $d$ satisfies $|(\psi_d)_{0,0}|^2\le(5\sqrt5-11)/2$; every parameter choice $d$ for which $|\alpha_P|=\sqrt{(\sqrt5-1)/2}$ for both $P$ satisfies $|(\psi_d)_{0,0}|^2=(5\sqrt5-11)/2$; and there exists a parameter choice $d$ for which both $|\alpha_P|=\sqrt{(\sqrt5-1)/2}$ for every $P$ and $|(\psi_d)_{0,0}|^2=(5\sqrt5-11)/2$.