option pricing - Jim Gatheral's ansatz

In the Ansatz section of Jim Gatheral's book Volatility Surface (page 32), he assumes

$$\mathbb E[x_s|x_T]=x_T\frac{\hat w_s}{\hat w_T}$$ where $\hat w_t:=\int_0^t \hat v_s ds$ is the expected total variance to time $t$, $\hat v_s$ is the unconditional expectation of the instantaneous variance $v_s$ at time $s$, $x_s = \log(S_s/K)$, and $T$ the time of the European option expiry.

Does anyone have a reference to a somewhat rigorous justification for this ansatz?

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As shown below in the comments, Quantuple found the answer by Peter Friz, Slade then provided a newer version.