= Solution
At time $t$, immediate exercise is worth $(S_t-K)^+$, while waiting is worth the <European call option> price with remaining maturity $\tau=T-t$. The conditional version of part (iii) gives
$$
C(t,S_t)\geq S_t-e^{-r\tau}K.
$$
Under the standard Black-Scholes assumption $r\geq0$, this is at least $S_t-K$, while a call value is nonnegative. Hence $C(t,S_t)\geq(S_t-K)^+$, and the investor should wait until $T$; with $\sigma>0$ and $\tau>0$, waiting has strictly positive time value.
If negative interest rates are allowed, the unconditional instruction “wait” need not be correct. The exact decision is to exercise when
$$
(S_t-K)^+\geq S_t\Phi(d_+(\tau))-Ke^{-r\tau}\Phi(d_-(\tau)),
$$
and otherwise to continue.
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