= Solution
The square root is concave on $[0,\infty)$, and $\sqrt{S_T}$ is <integrable> since $\mathbb E\sqrt{S_T}\leq\sqrt{\mathbb ES_T}$. For $t\leq T_1\leq T_2$, the <conditional Jensen inequality> and the <martingale> property give
$$
\mathbb E[\sqrt{S_{T_2}}\mid\mathcal F_{T_1}]
\leq\sqrt{\mathbb E[S_{T_2}\mid\mathcal F_{T_1}]}
=\sqrt{S_{T_1}}.
$$
Taking <conditional expectations> with respect to $\mathcal F_t$ and using the <tower property of conditional expectation> yields
$$
\boxed{C(t,T_2)\leq C(t,T_1)\quad\text{almost surely}.}
$$
Thus the maturity curve for a <square-root stock claim> is nonincreasing; it need not be strictly decreasing, as a constant stock gives equality. The inequality holds for every ordered pair of maturities. Under the usual right-continuous versions of the market processes, it gives the corresponding nonincreasing version of the maturity curve.
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