= Solution
For a deterministic local hidden variable, write the four possible measurement results as $q,r,s,t\in\{\pm1\}$. Its Bell combination is
$$
b=q(s-t)-r(s+t).
$$
If $s=t$, the first bracket vanishes and the second is $\pm2$; if $s=-t$, the second vanishes and the first is $\pm2$. Thus $b=\pm2$ for every hidden variable. Averaging over any local hidden-variable distribution gives $|\mathcal B|\leq2$, and in particular the requested upper bound.
For the specified settings, the correlation formula in part b gives
$$
\langle QS\rangle=\frac1{\sqrt2},\quad
\langle RS\rangle=\langle RT\rangle=-\frac{\cos\varphi}{\sqrt2},\quad
\langle QT\rangle=-\frac1{\sqrt2}.
$$
Consequently
$$
\boxed{\mathcal B=\sqrt2(1+\cos\varphi).}
$$
This violates the local bound precisely when
$$
\boxed{\cos\varphi>\sqrt2-1,\qquad
|\varphi-2\pi m|<\arccos(\sqrt2-1)\simeq1.14372\ \text{rad},\quad m\in\mathbb Z.}
$$
The maximum is $2\sqrt2$ at $\varphi=0$ modulo $2\pi$. The state remains maximally entangled at other <phases>, even when this fixed choice of real-plane analyser settings gives no violation. Local <phase> compensation or adapted measurement settings can recover the optimum.
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