Standard-imaginary strategy conjecture for the two-player Levine hats game

Let $\widehat{\mathcal{S}}_1$ be the class of standard-imaginary strategies and let $\widehat{\mathcal{M}}_1$ be the class of imaginary strategies used in the continuous Levine game. For measurable strategies $k$ and $g$ in $\widehat{\mathcal{S}}_1\times\widehat{\mathcal{M}}_1$, let $\varepsilon$ denote the success-indicator function. Standard-imaginary strategy conjecture.

$$V_2=\sup_{k,g\in\widehat{\mathcal{S}}_1\times\widehat{\mathcal{M}}_1}\int_{[0,1]^2}\varepsilon\left(2^{k(y)}x\right)\varepsilon\left(g(x)y\right)\,\operatorname{d}x\,\operatorname{d}y.$$

The conjecture asserts that restricting one player to a standard-imaginary strategy and the other to an imaginary strategy gives the exact optimal value for the two-player game. It is motivated by computational evidence that this restricted approximation does not exceed $7/20$, together with the fact that $7/20$ is attainable using $R_t$ strategies; no proof is given.