= Solution
The coefficient is $\boxed{k=1/8}$, but independence alone does not make the printed right-hand side $\mathcal F_t$-measurable. The future variance integral need not be known at time $t$. This is a genuine missing information assumption in the PDF.
The <Itô formula> or explicit <stochastic exponential> gives
$$
\frac{U_T}{U_t}
=\exp\left(\frac12\int_t^T\sigma_u\,dW_u
-\frac18\int_t^T\sigma_u^2\,du\right)
\exp\left(-\frac18\int_t^T\sigma_u^2\,du\right).
$$
If the volatility path is fixed at time zero and independent of $W$, conditioning on that path makes the first factor an exponential of a centered Gaussian variable with the compensating half-variance, so its conditional expectation is one. More generally, this conditioning works when enlarging the filtration by the entire independent volatility path preserves the Brownian property. For the usual joint filtration of Brownian history and independent volatility history, the valid <conditional square-root price under independent volatility> is
$$
\boxed{\mathbb E[U_T\mid\mathcal F_t]
=U_t\,\mathbb E\left[\exp\left(-\frac18\int_t^T\sigma_u^2du\right)\middle|\mathcal F_t\right].}
$$
When the integral is already $\mathcal F_t$-measurable, the outer conditional expectation can be removed, giving the intended printed formula. In particular this holds for deterministic volatility or an independent path disclosed initially.
For a counterexample to the unqualified printed assertion, take an independent fair Bernoulli variable $A\in\{0,1\}$, disclose it at time $1$, and let
$$
\sigma_u=A\min\{(u-1)^+,1\},\qquad S_0=1.
$$
With the filtration generated by the Brownian history and this disclosure, $W$ is Brownian and $\sigma$ is bounded, continuous, adapted, and independent of $W$. At $t=0,T=2$, the variance integral is $A/3$, while $\mathcal F_0$ is trivial. Direct Gaussian conditioning gives
$$
\mathbb E\sqrt{S_2}=\frac{1+e^{-1/24}}2,
$$
a constant. The proposed factor $e^{-A/24}$ is random, so cannot equal that conditional expectation. \b[The formula requires knowledge of future integrated variance; independence by itself is insufficient.]
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