Order estimates for Lebesgue constants of Fourier sums in Orlicz spaces

We consider the problem of order estimates for partial sums of trigonometric Fourier series as operators from Orlicz spaces $L^{\varphi}_{2\pi}$ to the space of $2\pi$-periodic continuous functions $C_{2\pi}$. It is established that an arbitrary function $\varphi$ generating an Orlicz class satisfies the estimate
$$ ||S_n(f)||_{C_{2\pi}} \le C \varphi ^{-1} (n) \ln (n+1) ||f||_{L^{\varphi}_{2\pi}}, \tag{*} $$
where $f \in L^{\varphi}_{2\pi}$, $n \in \mathbb{N}$, $S_n(f)$ is the $n$th partial sum of the trigonometric Fourier series of $f$, and the constant $C>0$ is independent of $f$ and $n$. In addition, it is shown that if the function $\varphi$ satisfies the $\Delta_2$-condition, then the estimate can be improved. More exactly,
$$ ||S_n(f)||_{C_{2\pi}} \le C \varphi ^{-1} (n) ||f||_{L^{\varphi}_{2\pi}}, \qquad f \in L^{\varphi}_{2\pi}, \, n \in \mathbb{N}, \, C=C(\varphi ). \tag {**} $$
Counterexamples are constructed, which show that if $\varphi$ satisfies the $\Delta_2$-condition, then estimate ($\ast \ast $) is unimprovable in order on the space $L^{\varphi}_{2\pi}$ and, if $\varphi$ satisfies the $\Delta^2$-condition, then estimate ($\ast $) is unimprovable in order on the space $ L^{\varphi}_{2\pi}$.